Written and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort. A separate exact scoped check covers the exactness, comma-category and small Kan value arguments, their needed prerequisite interfaces and four exercise solutions. It adds no whole-lesson or whole-course review. Original exposition uses CC0 1.0; linked complete proofs retain their stated terms.

Exactness through comma categories and small Kan values

Written and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original exposition: CC0. Linked complete proofs retain their stated terms.

A functor can preserve a universal object even when its source does not have every object of that kind. Comma categories give an exactness condition that makes sense before these existence questions are settled. Small cofinal witnesses then let us construct Kan values without asking the value category for arbitrary large colimits. We keep the maps of the universal cones and cocones throughout.

1. Conventions and retained complete proofs

Fix a Grothendieck universe \(\mathcal U\), ambient choice and the enlargement axiom in Universes and small categories, Sections 1–4. All category data are sets in the ambient setting. Categories are locally \(\mathcal U\)-small unless a larger ambient Hom universe is stated. A small category has both object and arrow sets \(\mathcal U\)-small. A finite category has finitely many objects and arrows, including the empty category.

We retain the following complete proofs with their hypotheses and actual maps.

• Compatible families and universal cones, Sections 2–3 and 6 constructs universal-family functors, chosen universal objects, pointwise limits, opposite transfer and the precise preservation comparisons.
• Filtered stages and finite limits, Sections 1 and 4 proves the ambient finite-diagram criterion for filteredness and the full small Set finite-filtered exchange, including the empty finite shape and naturality.
• One-sided fractions and saturation, Section 4 gives the generic outgoing Hom-density map and identity-class inverse, the existing-finite-colimit calculation and the finite-source converse.
• Universal tests for forks and finite limits, Section 6 and Universal forks and diagram constructions, Sections 8–9 give the terminal/product/equalizer and terminal/pullback tests, their duals and the full object/arrow presentations.
• Passing maps across an adjunction, Section 1 proves the universal-comma adjoint criterion, all Hom naturalities and its opposite. Fibres, slices and index adjunctions, Section 1 specializes it to the colimit/constant and constant/limit adjunctions.
• Change an index without changing its universal maps, Sections 1–5 proves cofinality through connected incoming commas, inverse cocone factors and both naturalities, ambient cofinal composition/cancellation, the small full image theorem and small-witness existence transfer.
• Limits and colimits of formal objects, Sections 1–2 proves the element filteredness witnesses and independent-coordinate filteredness with choice. Formal colimits and test objects, Sections 4 and 6 supplies arbitrary-base density, the identity-class inverse and universe descent.
• Hom, tensor and the exactness of module limits, Sections 2–5 proves the complete small filtered-module homology/exactness theorem and small product exactness over every associative unital ring.
• Extending diagrams by universal maps, Sections 3–4 proves the conditional left and right Kan constructions, their units and counits, both inverse adjunction factors and both-variable naturality. Extending a functor from its representables, Section 6 proves the opposite-Lan identification by inverse triple regrouping.
• Incoming comma tests and assembly from finite pieces, Propositions 3.2 and 5.1 supplies the complete small full filtered cofinal-witness proof. Its full source is a prerequisite for that use.

These interfaces are used at their exact scope below. For example, a proof about small Set colimits does not assert existence of the same colimit for every ambient index.

For \(F:C\to D\) and \(U\in D\), write

An outgoing comma arrow \(f:(X,u)\to(Y,w)\) satisfies \(wF(f)=u\). An incoming comma arrow \(f:(X,v)\to(Y,w)\) satisfies \(F(f)v=w\). Identities, composition and the faithful forgetful functors to \(C\) are the complete comma constructions in Zero maps, components and subobjects, Section 1. In particular

The arrow equations, as well as the objects, are reversed in (1.2).

2. Exactness before universal objects exist

Call \(F\) right exact if every \(K_U\) is filtered. Call it left exact if every \(H_U\) is cofiltered, equivalently if \(F^{\mathrm{op}}\) is right exact. Call it exact if both hold. These definitions include nonemptiness and impose no general finite-limit or finite-colimit existence assumption on \(C\).

Hom-density and its variance

The retained Section 4 proof in One-sided fractions gives the bijection

The full identity-class proof is retained there: the comma arrow \(a\) identifies a representative with the indicated identity representative of its image. The formula itself does not require filteredness. When its index is large, perform the quotient construction in one larger universe containing its entire data; the displayed target identifies its value with the given small Hom set.

Here are both naturality checks needed for the present use. For \(c:X'\to X\), the first-variable map sends \(a\) to \(ac\), and

For \(b:U\to U'\), the comma functor sends \((Y,u)\) to \((Y,bu)\), retaining its underlying arrows; the resulting class map satisfies \((bu)F(a)=b(uF(a))\). Thus (2.1), and hence its inverse, is natural in both variables.

Opposite transfer gives the correctly indexed incoming version

Its coefficient is contravariant on \(H_U\), so the colimit is over \(H_U^{\mathrm{op}}\). For \(c:X\to X'\), the class map is \(a\mapsto ca\) and \(F(ca)v=F(c)F(a)v\). For \(b:U'\to U\), the comma functor sends \(v\) to \(vb\) and \(F(a)(vb)=(F(a)v)b\). These give its two naturality squares explicitly.

Preserve an existing finite diagram

Theorem 2.1. A left exact functor preserves every specified existing finite limit, with its actual projections. A right exact functor preserves every specified existing finite colimit, with its actual coprojections.

Exact proof interface. Suppose \(L\), with \(p_j:L\to A(j)\), is a limit of a finite diagram \(A:J\to C\). Substitute its full Hom-family property into (2.3), then apply the retained finite-filtered Set exchange to the filtered category \(H_U^{\mathrm{op}}\). If necessary, use a larger universe for this entire Hom calculation. The resulting bijection is

Here \(\operatorname{Cone}(FA,U)\) denotes cones with vertex \(U\), as in Compatible families. The finite-exchange naturality and (2.3) show that these are the actual image components. Thus \(FL\) is a limit of \(FA\); target existence is a conclusion. No other finite limit in \(C\) was used. Reversing the retained proof gives \(b\mapsto(bF(c_j))_j\) for a specified finite colimit with legs \(c_j:A(j)\to Q\), so \(FQ\) with these image legs is its colimit. \(\square\)

If a target limit \(L'\) is independently chosen, its comparison \(\chi:FL\to L'\) satisfies \(p'_j\chi=F(p_j)\). For a target colimit \(Q'\), the comparison \(\psi:Q'\to FQ\) satisfies \(\psi c'_j=F(c_j)\). The theorem makes these precise maps invertible. An unrelated object isomorphism does not replace these equations. For the empty diagram, (2.4) asserts terminality, and the reversed equation asserts initiality.

Theorem 2.2. If \(C\) has finite limits, then left exactness is equivalent to preservation of finite limits. If \(C\) has finite colimits, then right exactness is equivalent to preservation of finite colimits.

Retained converse and scope binding. The complete converse in One-sided fractions, Section 4, forms finite colimits in \(K_U\): take a finite colimit in \(C\), apply preservation, and factor the compatible maps to \(U\). The factor gives its outgoing comma label and the original source coprojections are its comma coprojections. The initial object, binary coproducts and coequalizers therefore give all filtered witnesses. On opposites the same argument forms finite limits in \(H_U\), whose terminal object, products and equalizers give cofilteredness. The forward implication is Theorem 2.1. The converse uses existence of the finite operations in the source, including the empty one. \(\square\)

Consequently, with finite limits in \(C\), left exactness can be tested by preservation of a terminal object, binary products and categorical equalizers of parallel pairs. Equivalently, test a terminal object and pullbacks. Retain the full finite-construction proofs in Universal tests and Universal forks: products are pullbacks over the terminal object, and equalizers are pullbacks against a diagonal. Their comparisons retain both product projections or both pullback legs. With finite colimits in \(C\), the exact dual tests an initial object, binary coproducts and coequalizers, equivalently an initial object and pushouts. “Equalizer” here is the two-arrow categorical operation; it does not presume a zero morphism or additivity.

3. Adjoints, products and exact colimit functors

Universal comma objects

If \(F\) has a right adjoint \(G\), the retained adjoint theorem identifies \(K_U\) with the slice \(C/GU\). More precisely, the adjunction takes \(u:FX\to U\) to its unique adjoint arrow \(\bar u:X\to GU\); its naturality takes the outgoing triangle to the slice triangle. Thus

is terminal in \(K_U\). A category with a terminal object is filtered, by the retained three-axiom proof. Hence \(F\) is right exact. Its singleton terminal-object inclusion is also cofinal, so \(F\) is right small in the terminology of Section 6.

If \(F\) has a left adjoint \(L\), its unit gives an initial object \((LU,\eta_U:U\to FLU)\) in \(H_U\). It is cofiltered, and its singleton inclusion is co-initial. Thus \(F\) is left exact and left small. These conclusions need no finite operations in the source.

For each existing finite limit, the covariant Hom functor in its second variable preserves that limit by its full Hom-family property. If \(C\) has finite limits, \(\operatorname{Hom}_C(X,-)\) is therefore left exact by Theorem 2.2. If \(C\) has finite colimits, \(\operatorname{Hom}_C(-,Y):C^{\mathrm{op}}\to\mathsf{Set}_{\mathcal U}\) is left exact. If both kinds of finite operations exist, Hom is left exact in each argument separately.

If every \(I\)-colimit exists in \(C\), its chosen colimit functor is left adjoint to the constant-diagram functor, so it is right exact. If every \(I\)-limit exists, the chosen limit functor is right adjoint to constants, so it is left exact. Retain the complete chosen arrow actions and inverse factors in Fibres, slices and index adjunctions. Here existence of all diagrams of the stated index is an assumption, even when that index is ambient large.

A product of exact functors

Proposition 3.1. Let \(S\) be small and \(F_s:C_s\to D_s\) be left exact for every \(s\). Then \(\prod_sF_s\) is left exact. The right exact assertion holds with the same direction in every coordinate.

Proof of the comma binding. Fix \(U=(U_s)\). An object of the outgoing comma for the product is precisely a tuple \((X_s,u_s:F_sX_s\to U_s)\). Its arrow is a tuple \((f_s)\) with \(v_sF_s(f_s)=u_s\) for each \(s\). Sending this data to its coordinate tuple gives an isomorphism of whole categories

The inverse assembles the tuples and their arrows. Both functors preserve identities and compositions coordinate by coordinate, using the product-category proof in Products, disjoint unions and mixed functors, Section 1. The incoming version has the equations \(F_s(f_s)v_s=w_s\) and gives

Apply the retained independent-coordinate filteredness proof in Limits and colimits of formal objects, Section 2, in a larger universe containing all factors when necessary. Ambient choice selects the coordinate objects, common receivers and equalizing arrows together; the resulting tuples satisfy every required equation. A Hom set of either product category is a small-indexed product of small Hom sets, so its working-universe local bound is retained. For empty \(S\), both product categories and both relevant commas are terminal. This proves both assertions, including exactness when each factor is exact. \(\square\)

Set and module examples, with existence

For small filtered \(I\), Set has all \(I\)-colimits and finite limits. The complete canonical finite-filtered exchange proves that its colimit functor preserves finite limits. Theorem 2.2 gives left exactness; its right adjoint gives right exactness. Thus it is exact.

For small filtered \(I\), the module colimit functor over any associative unital ring \(R\) is also exact. Retain the full representative/equality and homology proof in Hom, tensor, Sections 3–4. Its categorical finite-limit consequence uses kernels and finite products; finite products of modules are finite direct sums, which the left-adjoint colimit functor preserves. The kernel comparison is the one proved by that homology calculation. Right exactness follows from the constant-diagram right adjoint.

For small discrete \(S\), retain the complete product short-exact-sequence proof in Hom, tensor, Section 5. Here is its categorical operation binding. Coordinate products preserve kernels, zero and finite products, hence finite direct sums. For a family \(f_s:M_s\to N_s\), apply the retained product proof to \(0\to\operatorname{im}f_s\to N_s\to\operatorname{coker}f_s\to0\). The map \(\prod_sM_s\to\prod_s\operatorname{im}f_s\) is surjective by the same choice-based lifting proof. Therefore the kernel of the original quotient map \(\prod_sN_s\to\prod_s\operatorname{coker}f_s\) is exactly the image of \(\prod_sf_s\); its quotient map is the actual cokernel. This identifies the canonical cokernel comparison, rather than just its object. Finite coproducts and these cokernels construct finite colimits: a parallel pair uses the cokernel of its difference. Theorem 2.2 gives both comma exactness conditions. The empty family gives the zero module and the zero short exact sequence. All carriers, products and indexed colimits here are small and actually exist.

The forgetful functor \(\mathsf{Mod}_R\to\mathsf{Set}_{\mathcal U}\) preserves finite limits, with the actual compatible tuples and equalizer subsets in Hom, tensor, Section 2, so it is left exact. It is not right exact: the initial zero module has a singleton underlying set, whereas the initial set is empty. The existing empty-colimit clause of Theorem 2.1 detects the failure.

Filteredness by itself does not produce every large-index colimit in \(\mathsf{Set}_{\mathcal U}\). For a precise boundary, let \(I=\operatorname{Fin}(\mathcal U)\), the ambient poset of finite subsets of the set \(\mathcal U\), and let \(D(A)=A\) with inclusion maps. The index is locally small and filtered: it contains the empty subset, unions give upper bounds and it is thin. Every value is small.

Suppose a small colimit \(Q\), with legs \(c_A\), existed. The singleton legs define \(q(u)=c_{\{u\}}(u)\). For distinct \(u,v\), there is a function \(\mathcal U\to2\) separating them, and its restrictions to the finite subsets form a cocone. Its factor through \(Q\) forces \(q(u)\ne q(v)\). Thus \(q\) embeds \(\mathcal U\) into the small set \(Q\), contrary to the complete Cantor and small-subset proof in Universes, Section 1. No such colimit exists. Valid filtered-colimit assertions must therefore specify a small index, an adequate cofinally small witness, or a stated larger value universe. The module assertion likewise requires actual indexed-colimit existence; it is not an unrestricted large-index conclusion.

4. Exact indexed colimits and base change

For a small category \(I\), suppose every \(I\)-colimit exists in \(C\). Say that \(I\)-colimits are exact if the actual colimit functor \(\operatorname{Fun}(I,C)\to C\) is exact in the comma sense. Say that small filtered colimits are exact if they exist and this holds for every small filtered \(I\).

Proposition 4.1. Suppose \(C\) has finite limits, every \(I\)-colimit exists, \(I\) is nonempty and connected, and \(I\)-colimits are exact. Then these colimits are stable under base change: for a cocone \(X_i\to Z\) and \(Y\to Z\), the canonical map

is invertible.

Scope and map binding of the retained argument. Connectedness says every cocone on a constant \(I\)-diagram has one common component; nonemptiness supplies a component. Thus its colimit is the constant value, with identity legs. In \(\operatorname{Fun}(I,C)\), take the pointwise pullback of \(X\to\Delta Z\leftarrow\Delta Y\). Left exactness of the colimit functor preserves this existing finite limit by Theorem 2.1. Its image vertices are \(\operatorname{colim}X\), \(Z\) and \(Y\), giving (4.1). This is the complete pullback argument retained in Dense probes and reconstruction from colimits, Section 4, with connectedness supplying its constant-diagram interface.

If \(c_i:X_i\to L\) are the chosen coprojections, the original \(i\)-leg of (4.1) is the pullback arrow whose two components are \(c_i\operatorname{pr}_X\) and \(\operatorname{pr}_Y\). This identifies the precise canonical comparison. A transformation of \(X\), or compatible arrows between \(Y\to Z\), makes its naturality square commute on each such leg and each pullback projection; the two universal uniqueness properties prove the square. \(\square\)

The full opposite assertion assumes finite colimits, all \(I^{\mathrm{op}}\)-limits, nonempty connected \(I^{\mathrm{op}}\) and exactness of that limit functor. It says those limits are stable under cobase change; the comparison is characterized by the pushout images of the original limit projections. All existence and map conditions reverse together.

The nonempty condition matters even when exactness holds. For empty \(I\) in modules, its colimit functor selects zero. Its outgoing and incoming commas are singleton categories, so it is exact. But base change over \(Z=0\) along a nonzero \(Y\to0\) would require the map \(0\to0\times_0Y=Y\) to be invertible, which fails.

5. Cofinality, filteredness transfer and elements

A functor \(\phi:J\to I\) is cofinal when each incoming \((i\downarrow\phi)\) is nonempty and connected. It is co-initial when each outgoing \((\phi\downarrow i)\) is nonempty and connected. Retain the whole ambient definition and inverse universal-family theorem in Change an index, Sections 1–2. A cofiltered or filtered category is connected by the common-source or common-receiver zigzag. Consequently a left exact functor is cofinal, and a right exact functor is co-initial. In particular a functor having a left adjoint is cofinal, and a functor having a right adjoint is co-initial. These claims do not assert that either index itself is filtered.

Transfer filteredness through outgoing commas

Theorem 5.1. If \(F:C\to D\) is right exact and \(D\) is filtered, then \(C\) is filtered. If \(F\) is left exact and \(D\) is cofiltered, then \(C\) is cofiltered.

Proof. Let \(A:J\to C\) be a finite diagram. The retained finite-cocone criterion supplies a cocone \(u_j:FA(j)\to U\) on \(FA\) in \(D\). Lift the entire diagram to \(K_U\) by sending \(j\) to \((A(j),u_j)\) and an arrow \(a:j\to k\) to \(A(a)\). Its comma equation is precisely

Filteredness of \(K_U\) supplies a cocone to an object \((X,v)\), with arrows \(r_j:A(j)\to X\) satisfying \(vF(r_j)=u_j\) and \(r_kA(a)=r_j\). Forget the comma label; these are a full \(C\)-cocone. For empty \(J\), the first step chooses an existing \(U\) because \(D\) is nonempty, and the second chooses an object of its nonempty outgoing comma. Thus every finite diagram, including the empty one, has a cocone. The criterion proves filteredness of \(C\).

For the dual, a finite diagram has a cone \(u_j:U\to FA(j)\) in cofiltered \(D\). It lifts to \(H_U\), with equation \(F(A(a))u_j=u_k\). A cone in this cofiltered comma has vertex \((X,v:U\to FX)\) and arrows \(r_j:X\to A(j)\) with \(F(r_j)v=u_j\) and \(A(a)r_j=r_k\). Forgetting gives the required cone, and the empty case again supplies nonemptiness. The opposite finite-diagram criterion proves cofilteredness. \(\square\)

Composition, on objects and arrows

Proposition 5.2. Right exact functors compose, and left exact functors compose.

Proof. Let \(C\xrightarrow F D\xrightarrow G E\), and fix \(W\in E\). There is a functor

The source triangle \(w'GFf=w\) is its target triangle; identities and composition follow from the laws of \(F\). At \((Y,v)\in(G\downarrow W)\), an outgoing comma object of \(P_W\) is exactly an object \((X,w)\) and an arrow \(a:FX\to Y\) with \(vG(a)=w\). Forgetting \(w\), which is determined by \(a\), gives

The inverse sends \((X,a)\) to \(((X,vG(a)),a)\). An arrow on either side is \(f:X\to X'\) with \(a'Ff=a\); that equation implies the original \(GF\)-comma triangle after applying \(vG(-)\). Both directions retain the same underlying arrows, so this is an isomorphism of entire categories. If \(F,G\) are right exact, \(P_W\) is right exact and its target is filtered. Theorem 5.1 makes its source filtered for every \(W\), proving the claim.

The incoming functor \(P_W^-:(W\downarrow GF)\to(W\downarrow G)\) has the same underlying \(F\)-arrow action. At \((Y,v:W\to GY)\), its incoming comma is isomorphic to \((Y\downarrow F)\): an arrow \(a:Y\to FX\) determines the source label \(G(a)v:W\to GFX\). Its arrow equation is \(Ff\,a=a'\), which also implies the larger incoming triangle. If \(F,G\) are left exact, this makes \(P_W^-\) left exact into a cofiltered target. The dual half of Theorem 5.1 proves the left assertion. \(\square\)

Finite exactness detected by elements

Let \(C\) have finite colimits, and let \(A:C^{\mathrm{op}}\to\mathsf{Set}_{\mathcal U}\). Its element category \(E_A\) has objects \((X,a)\), with \(a\in A(X)\), and arrows \(f:(X,a)\to(Y,b)\) satisfying \(A(f)b=a\).

Proposition 5.3. The functor \(A\) is left exact if and only if \(E_A\) is filtered. This assertion does not require \(E_A\) to be cofinally small.

Binding of the retained density and witness proofs. The complete arbitrary-base density theorem in Formal colimits, Section 4, gives

Its full inverse, element-leg naturality and universe descent are retained. If \(E_A\) is filtered, perform density and finite-filtered Set exchange in a larger universe containing its full data. Each represented coefficient takes a finite colimit of \(C\) to its Hom limit. The exchange therefore shows that the actual map from \(A\) of that colimit to the limit of its values is invertible. Its components are \(A(c_j)\), where \(c_j\) are the original coprojections; this is the density naturality check, not an arbitrary value isomorphism. Since \(C^{\mathrm{op}}\) has finite limits, Theorem 2.2 gives left exactness of \(A\). Its values and the resulting finite comparisons are still the original small sets.

Conversely, retain the complete witnesses in Limits and colimits of formal objects, Proposition 1.1. The initial object \(0\) has \(A(0)\) a singleton, giving a nonempty element category. The comparison \(A(X\amalg Y)\cong A(X)\times A(Y)\) gives a common receiver for any \((X,a),(Y,b)\). If \(f,g:(X,a)\rightrightarrows(Y,b)\), a coequalizer \(q:Y\to Q\) has \(A(Q)\) the equalizer of \(A(f),A(g)\); hence \(b=A(q)c\) for some \(c\). The element arrow \(q\) equalizes \(f,g\). These are all three filtered witnesses. \(\square\)

For comparison, the complete ind-recognition theorem in Ind-objects through their elements additionally requires cofinal smallness in order to obtain a small filtered presentation. That condition belongs to the presentation theorem and is not silently inserted into Proposition 5.3 or erased from ind-recognition.

The full covariance dual starts with finite limits in \(C\) and \(A:C\to\mathsf{Set}_{\mathcal U}\). Its covariant element arrows satisfy \(A(f)a=b\). Regard \(A\) as a presheaf on \(C^{\mathrm{op}}\); its presheaf element category is the opposite of this covariant one. The proposition says exactly that the covariant element category is cofiltered if and only if \(A\) is left exact. Its witnesses use a terminal object, binary products and equalizers, with the arrows pointing toward the original element objects.

6. Small functors and full cofinal witnesses

Call \(F:C\to D\) right small if every outgoing \(K_U\) is cofinally \(\mathcal U\)-small. Call it left small if \(F^{\mathrm{op}}\) is right small. Equivalently, every incoming \(H_U\) is co-cofinally small: its opposite has a small cofinal witness. Left smallness uses a small co-initial witness in \(H_U\), rather than a small cofinal witness there. Smallness does not assert filteredness or cofilteredness.

An essentially small source

Proposition 6.1. Every functor with essentially small source is both right small and left small.

Proof. Choose an equivalence \(R:B\to C\) with \(B\) small. For fixed \(U\), the outgoing comma \((FR\downarrow U)\) is small. Its object set is the small tagged union of \(\operatorname{Hom}_D(FRb,U)\) over \(b\in B\); target local smallness bounds each tag. Its arrows are subsets of source Hom sets \(\operatorname{Hom}_B(b,b')\), tagged over pairs of comma objects. The transported tagged-union bounds in Universes, Sections 2–3, therefore bound its whole arrow set. The same argument applies to \((U\downarrow FR)\), with \(\operatorname{Hom}_D(U,FRb)\).

The induced functor \((FR\downarrow U)\to(F\downarrow U)\) is fully faithful because \(R\) is: its lifted source arrows satisfy a triangle if and only if their images do. For \((X,u)\), choose an isomorphism \(r:Rb\to X\). The comma object \((b,uF(r))\) maps isomorphically to \((X,u)\) by \(r\), with inverse \(r^{-1}\). Thus this comma functor is an equivalence of entire categories. In the incoming case use \((b,F(r^{-1})v)\) for \((X,v)\), and the same isomorphism \(r\); its triangle is \(F(r)F(r^{-1})v=v\).

An equivalence is cofinal and co-initial, by the complete comma equivalence and connectedness criterion in Change an index. The two small comma categories therefore give the required outgoing cofinal and incoming co-initial witnesses. Empty source commas are small and their identity witnesses are valid. \(\square\)

Transfer cofinal smallness

Theorem 6.2. If \(F:C\to D\) is right small and \(D\) is cofinally small, then \(C\) is cofinally small. If \(F\) is left small and \(D\) is co-cofinally small, then \(C\) is co-cofinally small. The local Hom bounds on \(C,D\) are part of both statements.

Right-direction proof. By the retained small full image theorem, choose a small full cofinal \(S\subseteq D\). Form \(E=(F\downarrow S)\), whose objects are \((x,s,u:Fx\to s)\). An arrow \((h,a):(x,s,u)\to(y,t,v)\) has \(h:x\to y\), \(a:s\to t\), and

Identities and compositions have the two component laws; for consecutive arrows, substituting (6.1) gives the composite triangle. The functor \(q:E\to C\) forgets \(s,u,a\).

We first prove that \(q\) is cofinal. An object of \((x\downarrow q)\) is \((y,s,v,h:x\to y)\). It receives an arrow, with components \((h,1_s)\), from the canonical object \((x,s,vFh,1_x)\). The canonical objects and their arrows include an exact copy of \((Fx\downarrow S)\), sending \((s,b:Fx\to s)\) to \((x,s,b,1_x)\) and \(a\) to \((1_x,a)\). That category is nonempty and connected because \(S\to D\) is cofinal. Every object of \((x\downarrow q)\) is thus joined to this connected family. Hence \(q\) is cofinal. This proof uses complete paths, rather than only reaching some object of \(E\).

For every \(s\in S\), right smallness and the small full image theorem let us choose a small full cofinal subcategory \(A_s\subseteq(F\downarrow s)\). Ambient choice selects these witnesses for the small family \(S\). Let \(B\) be the full subcategory of \(E\) on all their tagged objects \((b,s,v: Fb\to s)\). Its object set is a small union of small sets. Each Hom set is a subset of \(\operatorname{Hom}_C(b,b')\times\operatorname{Hom}_S(s,t)\), determined by (6.1). The local bounds and tagged-union theorem give a small whole arrow set. Fullness retains every such arrow.

We prove \(B\to E\) cofinal. Fix \(e=(x,s,u)\). Cofinality of \(A_s\to(F\downarrow s)\) gives a receiving object \((b_s,v_s)\in A_s\) and \(h_s:x\to b_s\) with \(v_sFh_s=u\). This fixes an object \((b_s,(h_s,1_s))\) of \((e\downarrow B)\).

An arbitrary object of \((e\downarrow B)\) is an arrow \((h,a):e\to(b_t,t,v_t)\), with \(v_tFh=au\). It is therefore an object of the connected comma

computed inside \((F\downarrow t)\). Postcompose the fixed \((b_s,v_s)\) to \((b_s,av_s)\in(F\downarrow t)\). Since \(A_t\) is cofinal, choose a receiver \((z_t,w)\in A_t\) with \(k:b_s\to z_t\) and \(wFk=av_s\). Then \((z_t,w,kh_s)\) is a second object of (6.2), because \(wF(kh_s)=av_sFh_s=au\).

Every arrow of (6.2) gives an arrow of \((e\downarrow B)\) by retaining its underlying \(C\)-arrow and using \(1_t\) in \(S\). Thus its connected zigzag joins the arbitrary object to the arrow \((kh_s,a):e\to(z_t,t,w)\). Finally \((k,a):(b_s,s,v_s)\to(z_t,t,w)\) belongs to the full \(B\), and its composite with the fixed receiving arrow is exactly \((kh_s,a)\). This joins that zigzag to the fixed object. The comma is nonempty and connected, so \(B\to E\) is cofinal.

By the retained composition theorem, \(B\to E\to C\) is cofinal. Its source is small, giving cofinal smallness of \(C\). If desired, the small full image theorem turns this witness into a small full cofinal subcategory of \(C\). We did not assume that \(B\to C\) itself was full. If \(D\) is empty, \(F\) forces \(C\) empty and the empty witness proves the assertion directly. \(\square\)

Complete left-direction proof. Choose a small full co-initial \(S\subseteq D\) and put \(E^-=(S\downarrow F)\). Its objects are \((s,x,u:s\to Fx)\); an arrow \((a,h)\) satisfies \(Fh\,u=va\). At \(y\in C\), an object \(((s,x,u),h:x\to y)\) of \((q^-\downarrow y)\) maps to its canonical object \(((s,y,Fh\,u),1_y)\). Those canonical objects include the connected \((S\downarrow Fy)\). Hence \(q^-:E^-\to C\) is co-initial.

For each \(s\), choose a small full co-initial \(A_s\subseteq(s\downarrow F)\), and let \(B^-\) be full in \(E^-\) on their tagged objects. The same small unions and Hom-product subsets bound its entire data. For \(e=(s,x,u)\), choose a fixed \((b_s,v_s)\in A_s\) with \(h_s:b_s\to x\) and \(Fh_s\,v_s=u\). An arbitrary arrow \((a,h):(t,b_t,v_t)\to e\) has \(a:t\to s\) and \(Fh\,v_t=ua\). It is an object of the connected comma \((A_t\downarrow(x,ua))\) in \((t\downarrow F)\).

The object \((b_s,v_sa)\) of \((t\downarrow F)\) receives a chosen arrow \(k:z_t\to b_s\) from \((z_t,w)\in A_t\), with \(Fk\,w=v_sa\). Then \((z_t,w,h_sk)\) is another object of that connected comma, since \(F(h_sk)w=ua\). Its zigzag transfers into \((B^-\downarrow e)\). The arrow \((a,k):(t,z_t,w)\to(s,b_s,v_s)\) in the full \(B^-\), followed by the fixed arrow to \(e\), has components \((a,h_sk)\). Thus the zigzag connects the arbitrary object to the fixed one. This proves co-initiality of \(B^-\to E^-\), and co-initial composition makes its small source a witness for \(C\). The empty case reverses unchanged. \(\square\)

Smallness of composites and adjoint witnesses

Corollary 6.3. Right small functors compose; left small functors compose.

Proof. For right small \(F,G\), the functor \(P_W\) in (5.2) is right small by the entire outgoing-comma isomorphism (5.3). Its target \((G\downarrow W)\) is cofinally small. Both comma categories have Hom sets that are subsets of the appropriate locally small source Hom sets. Apply Theorem 6.2 to obtain cofinal smallness of \((GF\downarrow W)\), for every \(W\).

For the left assertion use \(P_W^-\) and its incoming-comma isomorphism \(((Y,v)\downarrow P_W^-)\cong(Y\downarrow F)\). It is left small into the co-cofinally small \((W\downarrow G)\). The complete left half of Theorem 6.2 gives the result, with the same local Hom bounds. \(\square\)

Section 3 already supplies the adjoint special cases: a right adjoint to \(F\) gives a terminal outgoing singleton cofinal witness; a left adjoint gives an initial incoming singleton co-initial witness. Exactness and smallness are therefore simultaneous consequences of the respective universal comma objects, with no essential-smallness hypothesis on \(C\).

7. Two Kan existence branches and finite exactness

Let \(\phi:J\to I\). The bases may be ambient large and are locally small. Restriction is \(\phi^*:\operatorname{Fun}(I,C)\to\operatorname{Fun}(J,C)\). Large bases can make transformation sets in these functor categories larger than \(\mathcal U\). Their comma exactness is then an ambient statement: use one larger universe containing the complete functor-category data for the Hom arguments in Section 2. This does not enlarge the colimit or limit hypothesis on \(C\); small cofinal witnesses in the original working universe supply the values below.

Left Kan values

Theorem 7.1. Either of the following assumptions supplies a left adjoint \(\operatorname{Lan}_\phi\) to \(\phi^*\).

1. \(\phi\) is right small and \(C\) admits all small colimits.
2. \(\phi\) is right exact and right small, and \(C\) admits small filtered colimits.

Under assumption 2, if \(C\) also has finite limits and its small filtered colimits are exact, then \(\operatorname{Lan}_\phi\) is exact.

Existence binding of the retained conditional Kan proof. For \(i\in I\), put \(K_i=(\phi\downarrow i)\), with forgetful \(r_i:K_i\to J\). Right smallness gives a small cofinal witness in \(K_i\). Under assumption 1, its restricted diagram \(\beta r_i\) has a small colimit. The complete cofinal existence theorem extends its cocone and gives the actual colimit over the entire \(K_i\).

Under assumption 2, \(K_i\) is also filtered and locally small. The complete small full filtered witness in Incoming comma tests, Propositions 3.2 and 5.1, gives a small full filtered \(S_i\subseteq K_i\) with cofinal inclusion. Its filtered colimit exists by assumption and transfers to the full comma. Thus both branches supply the pointwise objects required by the complete conditional Kan theorem, but only the second branch requires these commas to be filtered.

Retain that theorem's functor, inverse adjunction factors and both naturalities from Extending diagrams, Section 3. Its chosen pointwise value and original legs are

For \(a:i\to i'\), its arrow is the unique map with

For \(b:\beta\to\gamma\), the map of Kan values is characterized by \((\operatorname{Lan}_\phi b)_i\lambda^{\beta,i}_{(j,t)} =\lambda^{\gamma,i}_{(j,t)}b_j\). These formulas retain the actual whole-comma legs, regardless of which small witness computed the value. Universal uniqueness gives identities, composition and compatibility of the two arrow actions.

The unit is \(\eta^\beta_j=\lambda^{\beta,\phi j}_{(j,1_{\phi j})}\). The counit at \(D:I\to C\) has components determined by \(\varepsilon^D_i\lambda^{\phi^*D,i}_{(j,t)}=D(t)\). The retained adjunction bijection is the precise correspondence

Inserting the identity label recovers \(f_j\). Conversely, (7.2) and naturality of \(\alpha\) recover its component on every leg, so recover \(\alpha\). The full retained proof verifies the two triangle identities and naturality in both \(\beta,D\); the displayed formulas are its exact interface, rather than only an isomorphism of unspecified objects.

The finite Kan comparison

Proof of the exactness assertion. Its right adjoint \(\phi^*\) makes \(\operatorname{Lan}_\phi\) right exact by Section 3. This does not assume finite colimits in \(C\).

Let \(\beta:K\to\operatorname{Fun}(J,C)\) be a finite diagram, with pointwise limit \(L\) and original projections \(p_k:L\to\beta_k\). Pointwise existence, its arrow laws and projection naturality are the retained universal-family proof. At each \(i\), fix one small full filtered cofinal \(S_i\subseteq K_i\), independently of the finite diagram and all its vertices. Let \(j_s=r_i(s)\).

Exactness of the actual small filtered colimit functor for this \(S_i\), together with finite limits in \(C\), gives its canonical invertible comparison

This is Theorem 2.1 applied to that colimit functor. Both pointwise colimit identifications in (7.4) use the same cofinal witness and respect all original comma legs. Let \(M\) be the pointwise limit of \(\operatorname{Lan}_\phi\beta_k\), with projections \(\pi_k\). The actual cone \(\operatorname{Lan}_\phi(p_k)\) determines \(\chi:\operatorname{Lan}_\phi L\to M\). At \(i\), this canonical map satisfies, for every \((j,t)\in K_i\),

On the chosen \(S_i\) legs this is exactly (7.4), so \(\chi_i\) is invertible. Cofinal cocone extension ensures that (7.5) holds on the whole \(K_i\), not only selected representatives.

For completeness, its naturality in \(i\) can be tested without comparing the choices of \(S_i,S_{i'}\). For \(a:i\to i'\), compose the two candidate arrows from \((\operatorname{Lan}_\phi L)(i)\) to \(M(i')\) with \(\pi^{i'}_k\) and with an original input leg. The first route gives

The second route, using \(\chi_{i'}(\operatorname{Lan}_\phi L)(a)\), has the same last expression by (7.2) and (7.5). The limit projections and colimit legs determine their maps uniquely, so the square commutes. A transformation between two finite diagrams has induced limit map \(\ell:L\to L'\), with \(p'_{k,j}\ell_j=b_{k,j}p_{k,j}\). Testing its comparison square gives the common expression \(\lambda^{\beta'_k,i}_{(j,t)}b_{k,j}p_{k,j}\). The same uniqueness tests prove naturality in the finite diagram.

Empty finite \(K\) is included in (7.4): exactness of the small filtered colimit functor preserves the terminal object. The preceding tests then reduce to uniqueness of maps into that terminal object. Thus the whole actual image cone is a finite limit. Finite limits exist in \(\operatorname{Fun}(J,C)\), so Theorem 2.2 in the stated ambient Hom universe makes \(\operatorname{Lan}_\phi\) left exact. Together with right exactness, this proves exactness. \(\square\)

The full right Kan dual

There is a right adjoint \(\operatorname{Ran}_\phi\) to \(\phi^*\) under either of these distinct assumptions:

1. \(\phi\) is left small and \(C\) has all small limits.
2. \(\phi\) is left exact and left small, and \(C\) has small cofiltered limits.

Use incoming \(H_i=(i\downarrow\phi)\), which is co-cofinally small, and in the second branch cofiltered. The opposite small full witness theorem supplies a small full co-initial cofiltered subcategory. The retained cone existence transfer and conditional right Kan proof give

A transformation \(b:\beta\to\gamma\) satisfies \(\rho^{\gamma,i}_{(j,t)}(\operatorname{Ran}_\phi b)_i =b_j\rho^{\beta,i}_{(j,t)}\). Its counit at \(j\) is the identity-label projection \(\rho^{\beta,\phi j}_{(j,1_{\phi j})}\). For \(f:\phi^*D\to\beta\), the inverse adjunction factor \(\alpha:D\to\operatorname{Ran}_\phi\beta\) is characterized by

For \(\beta=\phi^*D\), the unit \(D\to\operatorname{Ran}_\phi\phi^*D\) therefore has projection \(D(t)\) at every incoming label. Retain all inverse factors, triangle identities and both naturalities from Extending diagrams, Section 4.

In branch 2, add exactness of small cofiltered limits and finite colimits in \(C\). Then \(\operatorname{Ran}_\phi\) is exact. Its left adjoint \(\phi^*\) gives left exactness without assuming finite limits in \(C\). For a finite diagram \(\beta_k\), with pointwise colimit \(Q\) and coprojections \(c_k:\beta_k\to Q\), one fixed small co-initial cofiltered witness at \(i\) gives the dual canonical exchange and an invertible map

If \(\iota_k^i\) are its source coprojections, its exact original-leg equations are

For \(a:i\to i'\), (7.7) makes both routes of its naturality square have projection/coprojection value \(c_{k,j}\rho^{\beta_k,i}_{(j,t'a)}\). For a transformation of finite diagrams, both routes give the same value by the induced colimit-map equation \(\ell_jc_{k,j}=c'_{k,j}b_{k,j}\). The universal tests prove both naturalities. The empty finite shape preserves the initial object. Finite colimits in \(\operatorname{Fun}(J,C)\) then give right exactness by Theorem 2.2. All working-universe and ambient Hom qualifications are the reversals of those above; no larger-universe limit hypothesis on \(C\) is introduced.

8. The exact extension from representables

Let \(F:J\to I\) have small bases, and suppose \(F\) is left exact. Write \(\widehat J=\operatorname{Fun}(J^{\mathrm{op}},\mathsf{Set}_{\mathcal U})\), and similarly for \(\widehat I\).

Corollary 8.1. The Yoneda extension \(\widehat F:\widehat J\to\widehat I\) is exact.

Retained opposite-Lan interface and specialization. For the correctly typed input \(A\in\widehat J\), the complete proof in Extending a functor from its representables, Section 6, naturally identifies

At \(V\in I\), its comma is \((F^{\mathrm{op}}\downarrow V)=(V\downarrow F)^{\mathrm{op}}\). The retained density expression for the left side has classes \([(X,a),u:V\to FX]\), with \(a\in A(X)\). The comparison regroups them as

Both sides impose exactly the same generating relation. The full inverse-class proof is retained there. Under \(b:A\to B\), both class maps replace \(a\) by \(b_X(a)\). Under \(r:V'\to V\), both replace \(u\) by \(ur\). This binds the two naturalities. At \(A=h_Y\), a represented triple \(a:X\to Y\) has represented value \(F(a)u:V\to FY\), so the identification also preserves the original represented legs.

The functor \(F^{\mathrm{op}}\) is right exact, and it is right small by Proposition 6.1. Set has finite limits and exact small filtered colimits. Apply the second branch and exactness assertion of Theorem 7.1 to obtain exactness of the right side of (8.1).

Exactness transfers through this natural isomorphism on whole commas: if \(\theta:H\to L\) is invertible, send an outgoing label \(u:HX\to U\) to \(u\theta_X^{-1}:LX\to U\), and an incoming label \(v:U\to HX\) to \(\theta_Xv:U\to LX\). Naturality preserves each triangle, and \(\theta^{-1}\) gives the inverse functors with unchanged underlying arrows. This proves the corollary for the actual extension, with its maps. \(\square\)

The covariance counterpart has a right exact \(F\) with small bases. Its extension between \(\operatorname{Fun}(J,\mathsf{Set}_{\mathcal U})\) and \(\operatorname{Fun}(I,\mathsf{Set}_{\mathcal U})\) is the exact \(\operatorname{Lan}_F\). It is Corollary 8.1 applied to \(F^{\mathrm{op}}\); the outgoing comma \((F\downarrow V)\) is filtered and its coefficient is the given covariant functor. This states the variance and value category explicitly.

9. Four graded exercises with full solutions

Exercise 1 — introductory: directions and empty finite diagrams

Let \(P=\{0<1\}\) and let \(T\) be the terminal category. Define \(f_0,f_1:T\to P\) by the indicated object. Compute all outgoing and incoming commas, exactness, smallness and the available adjoints. Identify the empty finite operation that the other exactness condition fails to preserve. Does \(f_0\times f_1\) have either exactness condition? Also explain the analogous empty-colimit obstruction for the module forgetful functor.

Solution. An outgoing comma object for \(f_0\) at \(i\) is an arrow \(0\to i\); there is exactly one for both \(i=0,1\). Both commas are terminal. Its incoming comma at \(0\) is terminal, and at \(1\) is empty because \(1\not\le0\). Thus \(f_0\) is right exact and fails left exactness.

For \(f_1\), the outgoing comma at \(0\) is empty and at \(1\) terminal. Both incoming commas are terminal, since \(i\le1\). Thus \(f_1\) is left exact and fails right exactness. All four types of comma are small, including the empty ones; the identity functors of these small commas are cofinal and co-initial. Hence both functors are right small and left small.

The constant \(r:P\to T\) is a right adjoint to \(f_0\): \(\operatorname{Hom}_P(0,i)\) and \(\operatorname{Hom}_T(*,r i)\) are the same singleton sets, naturally. It is a left adjoint to \(f_1\), since \(\operatorname{Hom}_P(i,1)\) is always a singleton. The opposite adjoints cannot exist, by the universal-comma criterion and the empty commas just computed.

The unique object of \(T\) is terminal and initial. Its image under \(f_0\) is not terminal in \(P\), so the existing empty limit is not preserved. Its image under \(f_1\) is not initial, so the existing empty colimit is not preserved. These identify the exact failures in Theorem 2.1; both sources have all finite operations.

For the product, (3.2) is terminal times the outgoing comma of \(f_1\); it is empty whenever the second target coordinate is \(0\). The incoming product comma is empty whenever the first coordinate is \(1\). Thus the mixed product has neither exactness condition. Proposition 3.1 requires the same condition in every coordinate.

Finally the zero module is initial, but its underlying singleton set is not initial in Set. The actual image of the empty cocone therefore fails universality. Products and equalizer subsets still give preservation of finite limits, so the forgetful functor is left exact. No additive interpretation of the categorical equalizer was used.

Exercise 2 — intermediate: transfer and the three element witnesses

(a) Prove the three filtered axioms for a source \(C\) when \(F:C\to D\) is right exact and \(D\) filtered, without assuming finite colimits in either category.

(b) For composable right exact \(F,G\), recover the full outgoing-comma isomorphism that proves exactness of the composite. State its incoming dual.

(c) Let \(C\) be the category of finite sets, and let \(A(X)=\operatorname{Map}(X,M)\) for a small set \(M\), possibly infinite. Give the three filtered witnesses in \(E_A\), identify the finite-colimit comparison of \(A\), and explain why a terminal element object need not exist. State the covariance dual precisely.

Solution. (a) Choose \(U\in D\); the nonempty \((F\downarrow U)\) supplies \(x\in C\). For \(x_1,x_2\), filteredness in \(D\) supplies \(u_a:Fx_a\to U\). Their outgoing comma objects have a common receiver \((z,v)\), giving arrows \(x_a\to z\) in \(C\).

For \(f,g:x\rightrightarrows y\), choose \(e:Fy\to U\) with \(eFf=eFg\). Put \(u=eFf\). Then \(f,g\) are parallel arrows from \((x,u)\) to \((y,e)\) in \((F\downarrow U)\). Its filteredness supplies a further comma arrow \(t:(y,e)\to(z,v)\) with \(tf=tg\). Its source equation is \(vFt=e\). Forgetting gives the required source equalizer witness. This proves all axioms, with no universal objects presumed. Replacing common receivers by common sources and postcomposition equalization by precomposition gives the full cofiltered statement as in Theorem 5.1.

(b) At \((Y,v:GY\to W)\), an object of \((P_W\downarrow(Y,v))\) is \(((X,w),a:FX\to Y)\) with \(w=vG(a)\). Send it to \((X,a)\). An arrow is exactly \(h:X\to X'\) with \(a'Fh=a\). The inverse inserts \(w=vG(a)\) and retains \(h\); applying \(vG(-)\) verifies the larger comma triangle. Thus both inverse functors preserve all identities and compositions. Their target is \((F\downarrow Y)\). Theorem 5.1 applies to \(P_W\) because \((G\downarrow W)\) is filtered.

For the incoming version, the label \(a:Y\to FX\) determines \(w=G(a)v\), and the arrow equation is \(Fh\,a=a'\). This identifies \(((Y,v)\downarrow P_W^-)\) with \((Y\downarrow F)\), giving the left exact composition by the cofiltered transfer theorem.

(c) An element object is a finite set \(X\) labelled by a function \(a:X\to M\); an arrow \(f\) satisfies \(bf=a\). The empty set and its unique function give nonemptiness. The disjoint union \(X\amalg Y\), labelled by \(a\amalg b\), receives both objects. If \(f,g:(X,a)\rightrightarrows(Y,b)\), the finite quotient \(q:Y\to Q\) generated by \(f(x)\sim g(x)\) has a unique descended function \(c:Q\to M\), because \(bf=bg\). Thus \(cq=b\), and \(q\) is the required element arrow equalizing \(f,g\).

For a finite colimit \(L\) with coprojections \(c_j\), its comparison sends \(a:L\to M\) to \((ac_j)_j\). The retained finite Set-colimit factors identify it bijectively with the compatible family of functions. Its variance is contravariant: it preserves finite limits in \(C^{\mathrm{op}}\), hence is left exact. When \(M\) is infinite, a terminal labelled finite object \((Y,b)\) would have to receive each singleton with label \(m\in M\). That forces \(b:Y\to M\) to be surjective, which is impossible for finite \(Y\). Filteredness did not require a terminal object.

The covariance dual says that for finite limits in \(C\), a covariant \(B:C\to\mathsf{Set}_{\mathcal U}\) is left exact exactly when the category with arrows \(B(f)b=b'\) is cofiltered. Its nonempty, common-source and parallel-pre-equalization witnesses use the terminal object, products and equalizers. It is a statement about actual finite limits, with no small-presentation conclusion.

Exercise 3 — advanced: small arrows, connected paths and a large index

(a) For the unique \(F:C\to T\), identify right/left exactness and right/left smallness. Give a small example which is both right small and left small but has neither exactness condition.

(b) In the proof of Theorem 6.2, fix \(e=(x,s,u)\), the receiving witness \((b_s,v_s,h_s)\), and an arbitrary outgoing witness \((b_t,v_t,h,a)\). Write the two equations making \(z_t\) connect that arbitrary witness to the fixed one. Give the complete reversed equations. Explain why the full arrow set, rather than only its objects, is small.

(c) Prove that \(\operatorname{Fin}(\mathcal U)\) is not cofinally small and that its inclusion diagram has no colimit in \(\mathsf{Set}_{\mathcal U}\). Contrast \(\operatorname{Fin}(M)\) for a small set \(M\), including empty \(M\).

Solution. (a) Both comma categories of \(F\) at the unique target are canonically \(C\), including every arrow. Hence right exactness means filteredness, left exactness means cofilteredness, right smallness means cofinal smallness, and left smallness means co-cofinal smallness. Take \(C\) to be the discrete two-object category. It is small, so its identity witnesses both smallness conditions. Its two objects have neither a common receiver nor a common source, so it has neither exactness condition. If \(C\) is empty it remains both kinds of small, but both exactness conditions fail by nonemptiness.

(b) The receiving equation is \(v_sFh_s=u\), while the arbitrary witness has \(v_tFh=au\). Choose \((z_t,w)\in A_t\) receiving \((b_s,av_s)\), with

Thus \(h\) and \(kh_s\) are objects of the connected \(((x,au)\downarrow A_t)\). Every arrow of its zigzag keeps its underlying source arrow and has target tag \(1_t\), so gives a zigzag in \((e\downarrow B)\). The arrow \((k,a):b_s\to z_t\) in the full \(B\) connects this path to the fixed witness, because its composite is \((kh_s,a)\). This establishes the required whole-comma connectedness; merely exhibiting a receiver would not do so.

For the reversal, the fixed equation is \(Fh_s\,v_s=u\). An arbitrary incoming witness has \(a:t\to s\) and \(Fh\,v_t=ua\). Choose \(k:z_t\to b_s\) with

The connected \((A_t\downarrow(x,ua))\) joins the arbitrary witness to \(h_sk\). The full-subcategory arrow \((a,k):z_t\to b_s\) then joins it to the fixed witness in \((B^-\downarrow e)\). These are the complete opposite path equations.

There is a small tagged union of selected objects over small \(S\). Each allowed arrow is a pair in a small \(C\)-Hom times a small \(S\)-Hom satisfying one triangle. The subset bound, then the small tagged union over all object pairs, bounds the whole arrow set. Choice selects the per-\(s\) witnesses and the coordinate lifts. Local source Hom bounds are needed even when there are few selected objects.

(c) The index is filtered and locally small by empty subsets, finite unions and thinness. If it had a small cofinal witness, the retained small full filtered-witness theorem would give a small receiving family \(S\) of finite subsets. Every singleton \(\{u\}\) would be contained in a member of \(S\), so \(\mathcal U=\bigcup_{A\in S}A\). The transported small-union bound would make this union small, contrary to the complete Cantor proof. This is precisely the receiving-family criterion, with its local Hom bound retained.

If its diagram \(A\mapsto A\) had a small colimit \(Q\), the singleton legs would give \(q:\mathcal U\to Q\). Restrict a separating function \(\mathcal U\to2\) to all finite subsets. Its unique colimit factor separates the two corresponding \(q\)-values. Thus \(q\) is injective, again making \(\mathcal U\) small. This rules out the literal arbitrary-large filtered Set-colimit claim.

For small \(M\), its power-set/subset bound makes \(\operatorname{Fin}(M)\) small, including its thin arrow set. The inclusion diagram has actual colimit \(M\). For a compatible family \(a_A:A\to Q\), define \(a(m)=a_{\{m\}}(m)\). Inclusion of the singleton into each containing finite subset shows \(a|_A=a_A\). Every element occurs in a singleton, proving uniqueness. If \(M\) is empty, the index has just \(\varnothing\) and the colimit is the empty set; the same factor statement holds. No enlargement of the value universe was required in this case.

Exercise 4 — challenge: why the two Kan branches differ

Let \(P=\{0<1\}\), \(T\) be terminal, and \(\phi_r:T\to P\) select \(r\in\{0,1\}\).

(a) Compute the left Kan extension of a set \(M\) along each \(\phi_r\), its unit, its counit on \(D=(D_0\to D_1)\), and its exactness. Explain which existence branch applies.

(b) Let \(C\) be the full category of nonempty small sets. Show it has small filtered colimits but that restriction along \(\phi_1\) has no left adjoint with these values. Identify the missing hypothesis.

(c) Compute both right Kan extensions in Set, their units and counits, and the finite comparison for the constant-valued case.

(d) Compute the presheaf Yoneda extensions of \(\phi_1\) and \(\phi_0\), including the direction of the presheaf arrow, the triple comparison and its two naturalities. Identify the exact one and the covariance counterpart.

Solution. (a) For \(\phi_1\), the outgoing comma at \(0\) is empty and at \(1\) singleton. Therefore

Its unit at the only source object is \(1_M\). Its counit to \(D\) is the unique map \(\varnothing\to D_0\) at \(0\) and \(1_{D_1}\) at \(1\). A transformation from (9.3) to \(D\) is exactly a map \(M\to D_1\); the naturality equation at \(0\to1\) is automatic. This gives the actual adjunction factors. Smallness follows from the finite bases, so the all-small-colimits branch applies. The filtered-only branch does not apply, because its outgoing comma at \(0\) is empty. The terminal source diagram \(M=1\) goes to a diagram with empty value at \(0\), rather than the terminal constant singleton diagram. Thus this left Kan functor is right exact by its adjoint and fails left exactness.

For \(\phi_0\), both outgoing commas are singleton: their labels are \(1_0\) and \(0\to1\). Hence

The unit is again identity. Its counit to \(D\) has components \(1_{D_0}\) and \(D(0\to1)\). A transformation from (9.4) to \(D\) is uniquely determined by \(M\to D_0\), with the other component its composite to \(D_1\). This is the retained adjunction formula on every original leg. Here \(\phi_0\) is right exact and right small; Set has exact small filtered colimits and finite limits, so the second branch proves exactness. Directly, (9.4) applies the identity functor in both coordinates and every pointwise finite comparison is identity on the corresponding finite Set limit.

(b) For a small filtered diagram of nonempty sets, the complete Set quotient model is nonempty: choose an index and an element of its value. Its carrier is small, and the full subcategory inherits its factors to nonempty targets. Thus \(C\) has those colimits. It has no initial object, since any nonempty \(A\) has at least two distinct constant maps to \(2\).

More strongly, suppose restriction at \(1\) had a left adjoint, and its value on a nonempty \(M\) were \((A\to B)\) with \(A,B\) nonempty. Take \(D=(2\to1)\) with its unique arrow. A transformation \((A\to B)\to D\) consists of any map \(A\to2\) and the unique \(B\to1\); every such pair is natural. There are at least two transformations. But its claimed adjunction would identify them with \(\operatorname{Map}(M,1)\), a singleton. This contradiction rules out the adjoint. The functor \(\phi_1\) is right small, but is not right exact. Right smallness alone with only filtered colimits cannot supply the empty comma value. The all-small-colimits branch also fails for this value category.

(c) Incoming commas for \(\phi_1\) are singleton at both objects, so \(\operatorname{Ran}_{\phi_1}M=(M\xrightarrow{1_M}M)\). Its counit is identity. The unit on \(D\) has components \(D(0\to1)\) and \(1_{D_1}\); the projection equations in (7.7) verify naturality. For \(\phi_0\), the incoming comma at \(0\) is singleton and at \(1\) empty. Thus \(\operatorname{Ran}_{\phi_0}M=(M\to1)\). Its counit is identity at \(0\), while its unit on \(D\) has components \(1_{D_0}\) and the unique \(D_1\to1\). Both use the all-small-limits branch. The latter functor fails to preserve the initial diagram, since \(M=\varnothing\) gives value \(1\) at \(1\); it is left exact by its left adjoint but not right exact.

The constant functors in (9.4) and \(\operatorname{Ran}_{\phi_1}\) preserve all existing pointwise finite operations. If \((M_k)\) is a finite diagram, their comparisons have value the identity on \(\lim_k M_k\) or on \(\operatorname{colim}_k M_k\), in both coordinates. The original projections or coprojections are unchanged, so the comparison is canonical and natural. Empty finite shapes give the constant singleton or constant empty diagram, respectively. This direct computation does not assert that every small cofiltered Set-limit functor is exact.

(d) A presheaf on \(P\) is an arrow with values \(A(1)\to A(0)\). For \(F=\phi_1\), its incoming commas at both objects are singleton. Therefore \((F^{\mathrm{op}}\downarrow V)=(V\downarrow F)^{\mathrm{op}}\) is singleton, and

The original density triples have the unique source object \(*\), an element \(m\in M\) and the unique arrow \(V\to1\). Regrouping \([(*,m),u]\leftrightarrow[(*,u),m]\) identifies each with \(m\). Under \(b:M\to N\), both routes send it to \(b(m)\); under \(0\to1\), both presheaf routes precompose the unique arrow and retain \(m\). These are the two naturality squares. The represented singleton has represented image \(h_1\), with a singleton at each value. Since \(\phi_1\) is left exact, Corollary 8.1 proves exactness, also visible directly in (9.5).

For \(F=\phi_0\), there is an incoming arrow \(V\to0\) only for \(V=0\). Hence its presheaf extension has \(M\) at \(0\), \(\varnothing\) at \(1\), and arrow \(\varnothing\to M\). The image of the terminal singleton presheaf fails to be terminal at \(1\), so it is not left exact. The hypothesis on the base functor cannot be dropped. Finally the covariance counterpart for the right exact \(\phi_0\) is exactly (9.4), an exact extension between covariant Set-functor categories. The difference in arrow direction is part of the comparison.

10. Proof provenance

The complete proofs named in Section 1 remain their original mathematical premises with their stated licensing. This lesson adds the product-comma binding, the generic filteredness/composition bridges, the complete smallness-transfer paths and the finite Kan comparison with its original maps. Its Set and module exchange, universal adjunction, cofinal factor and opposite-Yoneda-extension proofs are retained by those precise references. Kan extensions are also treated in Emily Riehl, Category Theory in Context, Chapter 6, and exact functors in Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Section 2.6.