Compatible families and universal cones

Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original text: CC0.

A diagram specifies values and the maps that relate them. A limit gathers data that agree along every map. A colimit assembles data while identifying what the diagram says should agree. Their universal properties explain why we can construct them one input at a time, interchange two indexing directions, and carry them through certain functors. They also explain the failures: an arbitrary function functor need not carry an assembly to the corresponding assembly of functions.

We use composition from right to left. Categories have ambient sets of objects and arrows; these may be larger than a chosen Grothendieck universe \(\mathcal U\). “Small” means \(\mathcal U\)-small, up to bijections. Assume ambient choice. Retain the complete size proofs in Universes and small categories, Sections 2–3 and 5, the complete functor-category and component-inverse proofs in Natural transformations, Sections 1–2, and the full Yoneda and Hom-encoding interfaces in Points and representations, Sections 1–2. We state a size condition whenever a construction is to stay in \(\mathcal U\).

An ordinary diagram is \(D:J\to C\). A projective system indexed by \(I\) is the same data written \(\beta:I^{\mathrm{op}}\to C\), with \(J=I^{\mathrm{op}}\). An inductive system is \(\alpha:I\to C\). Reversing this terminology does not reverse the target category.

1. Compatible families are a set

Let \(\beta:I^{\mathrm{op}}\to\mathsf{Set}\), and write \(s:i\to j\) for an arrow of \(I\). The constant singleton presheaf \(\Delta_1\) is terminal: from any presheaf there is exactly one family of functions to singletons, and every naturality equation has the unique such function on each side. A transformation \(\Delta_1\to\beta\) chooses elements \(x_i\in\beta(i)\); its naturality is precisely \(\beta(s)(x_j)=x_i\). Thus its transformation set is identified with

The identification evaluates a component at the singleton's point; its inverse makes that point's image \(x_i\). Both recover every component or coordinate. The maps \(p_i:L_\beta\to\beta(i)\) are coordinate projections. For empty \(I\), there is one empty tuple, one transformation, and no projections.

If the object set of \(I\) and each value of \(\beta\) are small, the retained indexed-product and subset proofs give a small \(L_\beta\). This does not require a small arrow set merely to cut out the subset. A small category has both size conditions, so in particular supplies the usual small-limit construction. If labels lie outside \(\mathcal U\), transport the values and index to universe members by those same proofs.

Proposition 1.1. For every set \(X\), there is a bijection, natural in \(X\) and \(\beta\),

Proof. The right side consists of functions \(f_i:X\to\beta(i)\) with \(\beta(s)f_j=f_i\). Define \(f(x)=(f_i(x))_i\). The equations put this tuple in \(L_\beta\). Projection recovers \(f_i\), and starting from \(f\) recovers every tuple \(f(x)\), so the constructions are inverse. For \(u:X'\to X\), both routes send \(x'\) to \((f_i(u(x')))_i\). For a transformation \(t:\beta\to\gamma\), the tuple map is \((x_i)\mapsto(t_i(x_i))\). It is compatible since \(\gamma(s)t_j=t_i\beta(s)\), and both routes in (1.2) give \((t_i f_i)_i\). Identity and composite transformations give identity and composite tuple maps coordinatewise. Empty \(X\), empty \(I\), and empty factors are included in these equations. \(\square\)

This proves that \(L_\beta\) is the object limit in sets, with its specified projections. For any \(\phi:J\to I\), restriction gives

An arrow \(t:j\to j'\) imposes the required equation because \(\phi(t)\) is an arrow of \(I\). Restricting along an identity changes nothing; successive restrictions select the same coordinates as the composite functor. The tuple map induced by a transformation commutes with restriction at each selected coordinate. Restriction can discard relations, so its direction matters.

2. Universal functors before universal objects

For \(D:J\to C\), a cone from \(T\) is a family \(a_j:T\to D(j)\) with \(D(s)a_j=a_{j'}\) for \(s:j\to j'\). A cocone to \(T\) is a family \(b_j:D(j)\to T\) with \(b_{j'}D(s)=b_j\). Define

Their literal values are ambient sets. The complete Hom-bifunctor proof in Products and mixed functors, Section 4 supplies both Hom variances. In its notation, \(P_D(T)=\lim_J\operatorname{Hom}_C(T,D(-))\) and \(Q_D(T)=\lim_{J^{\mathrm{op}}}\operatorname{Hom}_C(D(-),T)\). The compatible-family proof of Section 1, with the index renamed, supplies these limits of sets.

Precomposition sends a cone \(a_j\) to \(a_j u\) for \(u:T'\to T\). Postcomposition sends a cocone \(b_j\) to \(v b_j\) for \(v:T\to T'\). Associativity preserves each compatibility equation and both functor laws. Thus \(P_D:C^{\mathrm{op}}\to\mathsf{Set}\) is a presheaf, while \(Q_D:C\to\mathsf{Set}\) is covariant. Following the established conventions, \(P_D\) is an object of \(C^\wedge\), and \(Q_D\) an object of \(C^\vee=\operatorname{Fun}(C,\mathsf{Set})^{\mathrm{op}}\). The latter opposite affects arrows between these objects, not the covariant value functor \(Q_D\).

A limit object represents \(P_D\): it is an object \(L\) with a specified natural bijection \(\operatorname{Hom}_C(T,L)\simeq P_D(T)\). A colimit object represents \(Q_D\): it is \(K\) with a specified natural bijection \(\operatorname{Hom}_C(K,T)\simeq Q_D(T)\). Until such an object exists, the corresponding functor still exists. The symbols \(\lim D\) and \(\operatorname{colim}D\) name the objects when their representability is understood.

Proposition 2.1. The limit representation is equivalent to a cone \(p_j:L\to D(j)\) through which every cone factors uniquely. The colimit representation is equivalent to a cocone \(c_j:D(j)\to K\) through which every cocone factors uniquely.

Proof interface. Retain the full universal-element construction and naturality proof in Points and representations, Section 3, with \(A=P_D\). Its universal element is the image of \(1_L\), hence is the family \(p_j\). Its representation formula sends \(f:T\to L\) to \(P_D(f)(p)=(p_j f)_j\); its bijectivity is exactly existence and uniqueness of the factor. Conversely that factor property supplies the same natural representation. For \(Q_D\), use the proved covariant version: the image of \(1_K\) is \(c_j\), and \(f:K\to T\) maps to \((f c_j)_j\). These substitutions keep all representation maps and their endpoints. \(\square\)

The full uniqueness proof in that same section gives a unique isomorphism between two limits respecting every projection, and between two colimits respecting every coprojection. This includes empty diagrams: a limit of the empty diagram is terminal, and a colimit is initial, since their families are empty and their factor is unique.

A category admits \(J\)-limits if every \(J\)-diagram has a limit object; it admits \(J\)-colimits similarly. For projective-system notation use \(J=I^{\mathrm{op}}\). It admits finite or small limits when this holds for every finite or small category, respectively, and similarly for colimits. Here a finite category has finitely many arrows as well as objects. Having just finitely many objects is not this condition. Proposition 1.1 shows that the representability definition for sets recovers (1.1).

For \(t:D\to E\), a cone to \(D\) gives a cone to \(E\) by \(a_j\mapsto t_j a_j\). A cocone from \(E\) gives one from \(D\) by \(b_j\mapsto b_j t_j\). Naturality of \(t\) verifies each equation. Thus \(P_D\to P_E\), while the underlying covariant functors have \(Q_E\to Q_D\); the latter is an arrow \(Q_D\to Q_E\) in \(C^\vee\). When representatives exist, their arrows are

The universal properties give these arrows. For identities the identity satisfies their equations; for composites the composite arrows satisfy them. Uniqueness therefore proves both functor laws.

For \(\phi:H\to J\), restricting a cone or cocone gives \(P_D\to P_{D\phi}\) and \(Q_D\to Q_{D\phi}\) as value functors. The second direction becomes \(Q_{D\phi}\to Q_D\) in \(C^\vee\). With object representatives this says

The first arrow has projections \(p_{\phi(h)}\); the second has coprojections \(c_{\phi(h)}\). These families satisfy the restricted equations. Their identity, composition, and compatibility with a transformation of \(D\) follow by checking every indicated structural map and using uniqueness. In projective-system notation, the first arrow is \(\lim\beta\to\lim(\beta\phi^{\mathrm{op}})\), exactly (1.3).

3. Size changes and opposite categories

Suppose \(J\) is small and \(C\) is locally small in \(\mathcal U\). Each value in (2.1) is a subset of a small product of small Hom sets. The retained size proofs bound it even if the object set of \(C\) is larger than \(\mathcal U\). The complete Hom-encoding construction in Products and mixed functors, Section 4, supplies actual universe-member Hom values with their natural bijections; applying these coordinatewise gives encoded \(P_D,Q_D\). Different encodings have the same natural comparison after cancellation of their bijections. Their representation property is unchanged by that comparison.

Proposition 3.1. If \(\mathcal U\subseteq\mathcal V\), \(J\) is \(\mathcal U\)-small, and \(C\) is locally \(\mathcal U\)-small, existence of either object in Section 2 is unchanged by the enlargement. Its representative with structural maps is unchanged up to its unique compatible isomorphism.

Proof interface. Keep the \(\mathcal U\)-coded values of \(P_D,Q_D\) and regard them as \(\mathcal V\)-values. The family sets and maps are unchanged; alternative \(\mathcal V\)-codes have the natural comparisons just described. The complete full-faithfulness and representation-transfer proof in Points and representations, Section 7 now applies to \(P_D\). For \(Q_D\), apply it to presheaves on \(C^{\mathrm{op}}\). It lifts a representation and its inverse from the larger universe, or includes one from the smaller. The object category \(C\) stays fixed. The retained uniqueness theorem then compares the representatives and their families. No smallness of all the objects of \(C\) is inserted. \(\square\)

For \(D:J\to C\), the opposite diagram is \(D^{\mathrm{op}}:J^{\mathrm{op}}\to C^{\mathrm{op}}\). A cocone \(D(j)\to K\) becomes a cone \(K\to D(j)\) in the opposite category, with exactly the reversed compatibility equations. An arrow \(K\to T\) factoring a cocone becomes \(T\to K\) factoring that cone. The two factor properties, including uniqueness, are therefore identical. Consequently

when either side exists, and the dual formula follows by applying the same reversal again. This proves the result for arbitrary index categories in the stated ambient setting, in particular for a small discrete set of indices.

Calling \(D\) a projective system indexed by \(J^{\mathrm{op}}\) leaves it a diagram into \(C\), so its limit still uses incoming cones in \(C\). It does not become its colimit. For a discrete two-object diagram of singleton sets, the limit is a singleton and the colimit is a tagged two-element set. The product property and disjoint-union property follow respectively by assigning a pair of functions into the two sets and a function independently on each tagged summand. These objects are not even bijective.

4. Build a limit at every input

Theorem 4.1. Let \(J\) be any category. If \(C\) admits \(K\)-limits, then \(\operatorname{Fun}(J,C)\) admits them, computed at each \(j\in J\). Each evaluation functor preserves them. The same assertions hold for \(K\)-colimits.

All diagram collections here are ambient sets; use a larger universe for the functor category if necessary. Neither \(J\) nor \(K\) is required to be \(\mathcal U\)-small for this conditional theorem. If \(K\) is small and the values locally small, its preceding small-set formulation applies.

Proof. Let \(D:K\to\operatorname{Fun}(J,C)\). Choose at every ambient object \(j\) a limiting cone \(p_{k,j}:L(j)\to D(k)(j)\). For \(u:j\to j'\), the arrows \(D(k)(u)p_{k,j}\) form a cone to \(D(-)(j')\), because the maps of \(D\) are natural in \(J\). Define \(L(u)\) as its unique factor:

For an identity \(u\), \(1_{L(j)}\) satisfies (4.1). For \(u':j'\to j''\), substitution gives \(p_{k,j''}L(u')L(u)=D(k)(u'u)p_{k,j}\). Uniqueness gives \(L(1_j)=1\) and \(L(u'u)=L(u')L(u)\), so \(L\) is a functor. Equation (4.1) makes \(p_k:L\to D(k)\) natural. Their cone equations hold at each \(j\), hence as transformations.

For any cone \(a_k:T\to D(k)\), its values give unique \(h_j:T(j)\to L(j)\). For \(u:j\to j'\), both arrows \(L(u)h_j\) and \(h_{j'}T(u)\) have projection \(a_{k,j'}T(u)\), by (4.1) and naturality of \(a_k\). Uniqueness in the limit at \(j'\) makes them equal. Thus \(h:T\to L\) is natural; it factors the cone, and any other factor equals it at every \(j\). Evaluation has exactly the selected limiting cone, proving preservation.

For the dual assertion, use the explicit isomorphism \[ \operatorname{Fun}(J,C)^{\mathrm{op}} \cong\operatorname{Fun}(J^{\mathrm{op}},C^{\mathrm{op}}). \tag{4.2} \] It sends a functor to its opposite, and a reversed transformation to its componentwise opposite. Naturality reverses to the required equation; identities and composites reverse componentwise, and repeating the operation is its inverse. The complete functor-category laws retained above justify these operations. Since \(C^{\mathrm{op}}\) has \(K^{\mathrm{op}}\)-limits when \(C\) has \(K\)-colimits, the proved limit assertion applied to \(J^{\mathrm{op}}\) gives exactly the colimit assertion. Empty \(J\) has its unique diagram and transformation; the component proof is then vacuous and the universal property holds. Empty \(K\) gives pointwise terminal or initial objects when the hypothesis supplies them. \(\square\)

Here is the formal version promised in Section 2. Write \(h_X(T)=\operatorname{Hom}_C(T,X)\), and \(k_X\) for the object of \(C^\vee\) underlying \(q_X(T)=\operatorname{Hom}_C(X,T)\), with the existing encodings. For small \(\alpha:I\to C\), pointwise limits of sets exist in \(\operatorname{Fun}(C,\mathsf{Set})\), by Theorem 4.1 with the arbitrary domain \(C\). Thus a colimit of \(k\alpha\) exists in its opposite \(C^\vee\), and its underlying covariant functor is \(T\mapsto\lim_{I^{\mathrm{op}}}\operatorname{Hom}_C(\alpha(-),T)=Q_\alpha(T)\). Likewise the limit of \(h\beta\), for \(\beta:I^{\mathrm{op}}\to C\), exists in \(C^\wedge\) and is \(P_\beta\). The induced value maps agree with (2.1) by the Hom functor laws. Each formal object has a representative in \(C\) exactly when the corresponding object colimit or limit exists, by its definition. Existence of the formal object by itself does not supply its representative.

5. Two indexing directions

Retain the complete construction on functors and transformations in Natural transformations, Exercise 4: \(\operatorname{Fun}(I\times J,C)\cong \operatorname{Fun}(I,\operatorname{Fun}(J,C))\). It includes the mixed-arrow equation and both empty indices, so the inner diagrams and their arrow maps used below are already defined.

Theorem 5.1. If \(C\) admits both \(I\)-limits and \(J\)-limits, a diagram \(D:I\times J\to C\) has a limit, and

The isomorphisms are the unique ones compatible with every map to \(D(i,j)\), are natural in \(D\), and include empty indices. The full dual holds for colimits. In particular this applies to small \(I,J\); for a projective bifunctor use \(I^{\mathrm{op}},J^{\mathrm{op}}\).

Proof. Let \(E(i)=\lim_J D(i,-)\), with projections \(p_{i,j}\). The construction (2.2), or Theorem 4.1, gives \(E\) on arrows. The \(I\)-limit \(L=\lim_I E\) exists with projections \(r_i\). Put \(a_{i,j}=p_{i,j}r_i\). Naturality of \(p\) in \(i\) and the inner cone equations in \(j\) show compatibility in each coordinate. Every product arrow factors into the two separate-coordinate arrows, so these equations make \(a\) a cone to the whole product diagram.

Conversely, for a product cone \(b_{i,j}:T\to D(i,j)\), fix \(i\). Its unique inner factor is \(t_i:T\to E(i)\). For \(f:i\to i'\), the two arrows \(E(f)t_i\) and \(t_{i'}\) have the same \(j\)-projections \(b_{i',j}\); hence they agree. The \(t_i\) form an \(I\)-cone, and give a unique \(t:T\to L\). It has \(a_{i,j}t=b_{i,j}\). Any factor with these equations has \(r_i t=t_i\) by inner uniqueness and is then \(t\) by outer uniqueness. This proves the product universal property without assuming product-index limits in advance.

Interchanging \(I,J\) repeats this proof and gives the same universal cone. The full compatible-uniqueness interface of Section 2 yields (5.1). For a transformation \(D\to D'\), all three induced limit maps have the same composites to every \(D'(i,j)\); uniqueness makes the comparisons commute with that map. Identity comparisons and successive changes of order agree for the same reason: all respect the specified product projections. If an index is empty, the cone to its empty product diagram is empty, and the same two-stage factor proof still gives its unique factor; no object of an empty index was chosen. Apply the proved argument in \(C^{\mathrm{op}}\), with \(I^{\mathrm{op}},J^{\mathrm{op}}\), to obtain all colimit assertions and their coprojection compatibility. \(\square\)

6. When a functor keeps the universal property

For \(F:C\to C'\) and a limit cone \(p_j:L\to D(j)\), its image is a cone \(F(p_j):F(L)\to FD(j)\). If the image diagram has a limit \(L'\), it determines a canonical comparison \(F(L)\to L'\). For a colimit \(c_j:D(j)\to K\), its image gives the canonical comparison \(K'\to F(K)\) from an image-diagram colimit. These are characterized by

We say that \(F\) preserves the specified diagram limit or colimit when its image cone or cocone is universal. This includes existence of the corresponding object in \(C'\). If \(L'\) or \(K'\) exists, it is equivalent to invertibility of the appropriate map in (6.1), by compatible uniqueness and transport of the factor property along an isomorphism. Having an arbitrary object isomorphism without these equations is not this test. Preservation of \(J\)-limits or \(J\)-colimits means this for every such diagram, when they all exist in \(C\).

Even without object representability in \(C'\), the cone induces \(h_{F(L)}\to P_{FD}\) in \(C'^\wedge\): at \(T\), send \(a:T\to F(L)\) to \((F(p_j)a)_j\). The cocone induces \(q_{F(K)}\to Q_{FD}\) as underlying covariant functors, sending \(a:F(K)\to T\) to \((aF(c_j))_j\); this is \(Q_{FD}\to k_{F(K)}\) in \(C'^\vee\). Their naturality follows by composition. Such a comparison is an isomorphism precisely when its component maps give the complete universal factor property, and therefore the representative. These are the formal comparisons behind (6.1).

Theorem 6.1 (retained adjoint preservation). A right adjoint preserves every existing diagram limit; a left adjoint preserves every existing diagram colimit. Existence of all limits or colimits in the target category is not an additional assumption.

Exact proof interface. Retain [Stacks, Adjoints, lemma-adjoint-exact], including its complete proof and formal dual. With \(F\) a right adjoint and \(G\) its left adjoint, specialize its universal Hom test at \(T\in C'\). The adjunction identifies maps \(T\to F(L)\) with maps \(G(T)\to L\); the latter correspond to compatible families \(G(T)\to D(j)\). Componentwise adjunction identifies these with compatible families \(T\to FD(j)\). Compatibility is justified by adjunction naturality, provided by the complete correspondence in Passing maps across an adjunction, Section 1. That naturality also identifies the image of \(a:T\to F(L)\) as \((F(p_j)a)_j\). Thus the retained theorem proves exactly the canonical cone's property, including its representative. For a left adjoint use the same cited theorem on maps out of its image colimit; the component maps are exactly \(F(c_j)\). This is a specialization and a comparison-map check of the existing proof, rather than a new proof of adjoint preservation. \(\square\)

7. Linear maps and point functors

Let \(k\) be any field, choose \(\mathcal U\) large enough to contain its data, and let \(V\) be any small \(k\)-vector space. The category \(k\text{-}\mathsf{Vect}\) of small vector spaces has all small limits and colimits by the complete coordinate-product and direct-sum/relations constructions in Module limits, Section 2 and Kernels and cokernels, Section 5.

The exact arbitrary-module tensor adjunction in Passing maps, Section 5.2, specialized to \(R=k\) and \(K=V\), is \(-\otimes_k V\dashv\operatorname{Hom}_k(V,-)\). Retain its complete balance, inverse, linearity, and naturality checks. By Theorem 6.1, \(\operatorname{Hom}_k(V,-)\), valued in vector spaces, preserves all small limits, without a finite-dimensional hypothesis.

Proposition 7.1. This Hom functor preserves all small colimits exactly when \(V\) is finite-dimensional.

Proof. For finite dimension \(n\), choose a basis \(e_1,\ldots,e_n\) and its dual basis \(e^1,\ldots,e^n\). For any \(W\), define

The second rule is bilinear and balanced, so gives a linear map from the tensor product. Applied to the first rule at \(x\), it gives \(\sum_r e^r(x)f(e_r)=f(x)\). Applied in the reverse order to \(w\otimes\ell\), it gives \(\sum_r\ell(e_r)w\otimes e^r=w\otimes\ell\), because \(\ell=\sum_r\ell(e_r)e^r\). Pure tensors generate the tensor product, proving both inverse equations on every element. A linear map \(W\to W'\) commutes with the displayed formulas, so this is a natural isomorphism in \(W\). At \(n=0\), both sides are zero and the empty sum gives the same inverse assertions.

The functor \(-\otimes_k V^*\) is a left adjoint by the same complete tensor adjunction. It preserves all small colimits by Theorem 6.1. Transporting its universal cocones through the natural isomorphism (7.1) proves preservation by Hom: naturality makes every image coprojection agree under that transport. This concerns all small colimits, not only filtered ones.

Now let \(V\) have infinite dimension. Its finite-dimensional subspaces form a small set \(\mathcal F\): it is a subset of the power set of the small underlying \(V\). Ordered by inclusion, \(\mathcal F\) is a small category, nonempty because it contains zero, and filtered because \(U+U'\) is a finite-dimensional common upper bound. There are no distinct parallel arrows to equalize. The inclusion diagram has colimit \(V\). To check this property, compatible linear maps \(f_U:U\to W\) define \(f(v)\) in any subspace containing \(v\). Two choices lie in their sum, so give the same value. A subspace containing \(v,v'\) proves additivity there; one containing \(v\) proves scalar compatibility. Restriction recovers every \(f_U\), and these subspaces cover \(V\), proving uniqueness.

The maps \(\operatorname{Hom}_k(V,U)\to\operatorname{Hom}_k(V,U')\) are injective postcomposition maps. The colimit of this diagram is their directed union inside \(\operatorname{End}_k(V)\). Indeed a finite sum of maps landing in finite-dimensional subspaces lands in their sum, so the union is a vector subspace. Any compatible family of linear maps out of the stage spaces defines a unique linear map on the union; equality can be checked in a common stage. Its image under the canonical colimit comparison consists exactly of endomorphisms with finite-dimensional image. Every stage map has that property, and any such endomorphism factors through its image, one of the stages. The identity of \(V\) does not have that property. Thus this comparison is not surjective and cannot be an isomorphism. \(\square\)

Yoneda gives a different example. If \(D:J\to C\) has limit \(L\), then \(h_L(T)\simeq\lim_J h_{D(j)}(T)\) with the map \(f\mapsto(p_j f)_j\), by Section 2. These bijections are natural in \(T\), and Theorem 4.1 computes the presheaf limit pointwise, so \(h:C\to C^\wedge\) preserves every existing \(J\)-limit for which these presheaves and pointwise limits are formed.

It generally fails to preserve colimits. In sets, take the two singleton objects whose coproduct is \(A=\{a,b\}\). The presheaf coproduct \(h_1\sqcup h_1\) is pointwise, by Theorem 4.1. At \(T=\{0,1\}\), its value has two elements, one tagged constant function for each summand. The canonical comparison to \(h_A(T)\), a four-element function set, has exactly the two constant functions as its image. It misses both nonconstant functions, so the comparison is not invertible. All objects here can be taken in the chosen universe.

8. Constant diagrams and the whole identity

The complete component proof in Products and mixed functors, Exercise 3 identifies transformations between constants with one underlying arrow per zigzag component: \[ \begin{gathered} \operatorname{Nat}(\Delta_X,\Delta_Y)\\ \simeq\prod_{\pi_0(I)}\operatorname{Hom}_C(X,Y). \end{gathered} \tag{8.1} \] Its proof uses only identity arrows of the constants and equality propagated along zigzags, so applies to any ambient \(C\). Retain its forward and inverse assignments, including the empty product, and the complete zigzag construction in Zero maps, Section 4.

If \(I\) is nonempty and connected, (8.1) has a single factor. A cone from \(T\) to \(\Delta_X\) is exactly a transformation \(\Delta_T\to\Delta_X\), hence one map \(T\to X\). A cocone from \(\Delta_X\) to \(T\) is one map \(X\to T\). The bijections commute with composition of those maps by the retained component formulas, so they are the representations in Section 2. Both object limits are \(X\), with every structural map \(1_X\). The empty index instead gives a terminal or initial object; it does not make every \(X\) universal. Nonempty connectedness is the condition used here.

Proposition 8.1. Let the identity diagram \(1_C:C\to C\) have a colimit represented by \(S\). Then \(S\) is terminal.

Here the index is \(C\) itself. This is a conditional representability assertion in the ambient setting, even when \(C\) is not \(\mathcal U\)-small. It does not assert that the corresponding family functor takes small values.

Proof. Write its universal cocone \(a_X:X\to S\). For every \(f:X\to Y\), compatibility says \(a_Y f=a_X\). Its universal property detects parallel arrows out of \(S\): if \(u,u':S\to Z\) satisfy \(u a_X=u'a_X\) for all \(X\), they factor the same cocone, so are equal. Apply compatibility to \(a_X:X\to S\). It gives \(a_S a_X=a_X\) for every \(X\), hence this detection gives \(a_S=1_S\). If \(f:X\to S\) is any arrow, compatibility again gives \(a_S f=a_X\), so \(f=a_X\). There is therefore exactly one arrow \(X\to S\) for every \(X\), which is terminality. The representing object already guarantees that \(C\) is nonempty. \(\square\)

9. Attach a compatible element

Retain the full construction in Points and representations, Section 3: for a presheaf \(A:C^{\mathrm{op}}\to\mathsf{Set}\), its element category \(E_A\) has objects \((X,x\in A(X))\) and arrows \(f:(X,x)\to(Y,y)\) with \(A(f)(y)=x\). Its identities and composition, ambient sets, and local smallness have already been proved there.

Theorem 9.1. Suppose \(C\) admits \(I\)-colimits, and for every \(D:I\to C\), the canonical map \[ \begin{gathered} A(\operatorname{colim}_I D) \longrightarrow\lim_{I^{\mathrm{op}}}A(D(-)),\\ x\longmapsto(A(c_i)(x))_i \end{gathered} \tag{9.1} \] is bijective. Then \(E_A\) admits \(I\)-colimits computed on the underlying objects, and its projection to \(C\) preserves them. The index can be any ambient category for which these conditions hold.

Exact proof interface. Retain the complete element-colimit argument in Solution sets and universal representations, Section 1. Its construction takes one diagram \((X_i,x_i)\), forms its underlying colimit \(L\), and uses the preservation bijection to supply its unique element \(x\). For an arbitrary cocone to \((T,t)\), the underlying unique factor \(b:L\to T\) has \(A(b)(t)=x\), as proved there by comparison after every \(A(c_i)\) and injectivity of the same bijection. Its uniqueness as an element-category arrow is also included. Specialize that argument with the underlying colimit supplied by our fixed \(I\)-hypothesis and the bijection supplied by (9.1). The argument uses no colimits of other shapes and no smallness beyond the existence of these ambient family sets. Thus it applies to the stated fixed, possibly larger index as well. Its empty-diagram argument supplies the unique element on the initial object. The retained construction computes the chosen underlying cocone, so its projection preserves the colimit. \(\square\)

For \(F:C\to C'\) preserving \(I\)-colimits and \(Y\in C'\), set \(A(X)=\operatorname{Hom}_{C'}(F(X),Y)\). The complete Hom variance interface makes this a presheaf. Preservation and the colimit universal Hom property identify (9.1) with a bijection: a map \(F(L)\to Y\) corresponds to the compatible family \(F(X_i)\to Y\). The naturality of that correspondence gives exactly its displayed component map. The element category is \((F\downarrow Y)\), with identical objects, arrows, and compositions, by the full comma construction in Zero maps, Section 1. Theorem 9.1 therefore proves the result for this general comma category. For \(F=1_C\), it retains the existing slice-colimit proof in Initial objects and universal colimits, Section 1; no restriction to this special case was used.

10. Natural transformations as one compatible family

Define \(\mathsf{Fac}(C)\), also denoted \(\operatorname{Mor}_0(C)\), to have the arrows \(f:X\to Y\) of \(C\) as objects. An arrow from \(f\) to \(g:X'\to Y'\) is a pair

The second side is reversed. This construction differs from the ordinary square category in Zero maps, Section 1, whose second side goes \(Y\to Y'\).

The identity is \((1_X,1_Y)\). If \((u,v):f\to g\) and \((u',v'):g\to h:X''\to Y''\), their composite is \((u'u,vv')\), since \(f=v(v'hu')u=(vv')h(u'u)\). The first coordinate uses composition in \(C\), the second composition in \(C^{\mathrm{op}}\). For a third pair, either parenthesization gives \((u''u'u,vv'v'')\); both units reduce each coordinate to itself. This proves every category law, including its reversed side.

Its object set is the ambient arrow set of \(C\), and each Hom is a subset of \(\operatorname{Hom}_C(X,X')\times\operatorname{Hom}_C(Y',Y)\); tagged union bounds the ambient total arrow set. The retained product/subset proofs show local smallness when \(C\) is locally small. If \(C\) is a small category, its arrow set and these small indexed Hom sets also show that \(\mathsf{Fac}(C)\) is small.

For \(\alpha,\beta:I\to C\), define a presheaf \(H_{\alpha,\beta}\) on \(\mathsf{Fac}(I)\) by

For an arrow \(f\to g\) in (10.1), its function goes from the value at \(g\) to the value at \(f\), with \(h:\alpha(i')\to\beta(j')\). The result indeed goes \(\alpha(i)\to\beta(j)\). Identities give \(h\). For a successive arrow \((u',v'):g\to l\), the composite action gives \(\beta(v)\beta(v')h\alpha(u')\alpha(u) =\beta(vv')h\alpha(u'u)\), which is the action of its composite, in the required presheaf order.

Theorem 10.1. There is a bijection, contravariantly natural in \(\alpha\) and covariantly natural in \(\beta\),

This is an ambient-set statement for any ambient \(I,C\). For small \(I\) and locally small \(C\), its compatible-family sets and encoded values are small by the preceding size arguments.

Proof. Naturality gives the equality of the two formulas for \(h_f\). For \((u,v):f\to g\), let \(g:i'\to j'\), so \(f=vgu\). Then \[ \begin{aligned} \beta(v)h_g\alpha(u) &=\beta(v)\beta(g)\eta_{i'}\alpha(u)\\ &=\beta(v)\beta(g)\beta(u)\eta_i\\ &=\beta(f)\eta_i=h_f. \end{aligned} \tag{10.4} \] The family is compatible with every factorization arrow. Conversely, given a compatible family \(h_f\), put \(\eta_i=h_{1_i}\). For each \(f:i\to j\), there are arrows \(f\to1_i\) given by \((1_i,f)\) and \(f\to1_j\) given by \((f,1_j)\). Their two compatibility equations are \[ h_f=\beta(f)\eta_i=\eta_j\alpha(f). \tag{10.5} \] They prove naturality of \(\eta\). Starting with \(\eta\) recovers it at every identity; starting with the family recovers every \(h_f\) by (10.5). These are both inverse checks.

For \(a:\alpha'\to\alpha\) and \(b:\beta\to\beta'\), send a value \(h\) at \(f:i\to j\) to \(b_j h a_i\). This commutes with (10.2): naturality gives \(\beta'(v)b_{j'}=b_j\beta(v)\) and \(a_{i'}\alpha'(u)=\alpha(u)a_i\), so either route gives \(b_j\beta(v)h\alpha(u)a_i\). It is therefore a transformation of the presheaves. At a family coming from \(\eta\), naturality of \(b\) identifies its \(f\)-value with \(\beta'(f)b_i\eta_i a_i\), the family of \(b\eta a\). This proves both claimed variances and the naturality of the inverse identity evaluation. If \(I\) is empty, its factorization category is empty too, and both sides of (10.3) are the singleton of empty data. \(\square\)

11. Four graded exercises with full solutions

Exercise 1 (introductory: an idempotent collects and identifies)

Let the one-object category \(\mathsf{Pr}\) have arrows \(1,p\) with \(p^2=p\). On \(S=\{0,1,2,3,4,5\}\), let \(e\) fix \(0,3,5\) and send \(1\mapsto0\), \(2\mapsto3\), \(4\mapsto5\). Compute the limit and colimit of the functor \(D:\mathsf{Pr}\to\mathsf{Set}\) with \(D(p)=e\), and prove their universal properties. For the functor from the one-object identity category selecting the object of \(\mathsf{Pr}\), compute both restriction comparisons from Section 2.

Solution. Each image of \(e\) is fixed, so \(e^2=e\). The category and functor laws are exactly those already proved in Building categories, Section 4. A cone is a function \(f:T\to S\) with \(ef=f\). Thus its values lie in \(L=\{0,3,5\}\), and it has a unique factor \(T\to L\) through inclusion. The inclusion is the specified limiting projection.

A cocone is a function \(g:S\to T\) with \(ge=g\). It is constant on each of \(\{0,1\},\{2,3\},\{4,5\}\). Conversely constancy on these three classes makes \(ge=g\). Let \(Q\) be the set of those classes and \(q:S\to Q\) their quotient function. Then \(g\) defines a unique map \(Q\to T\) taking a class to its common value, so \(q\) is the colimit coprojection. The map \(Q\to L\) sending a class to its fixed representative is a bijection. It happens to identify the two underlying objects in this example, but their structural maps and factor properties have the opposite directions.

After restricting to the identity-only category, both object limits are \(S\): a one-object cone or cocone is just its single function, factored through \(1_S\). The limit comparison is \(L\hookrightarrow S\), and the colimit comparison is \(S\xrightarrow{q}Q\), by their structural equations. Their sizes are respectively \(3\to6\) and \(6\to3\), showing concretely the two directions and their failure to be isomorphisms.

Exercise 2 (intermediate: two independent equalizer conditions)

Let \(I,J\) each have two objects and two parallel arrows \(0\rightrightarrows1\). Define a diagram on \(I\times J\) with values \(S=\{0,1,2,3,4,5\}\) at \((0,0)\), \(A=B=\{0,1\}\) at \((1,0),(0,1)\), and a singleton at \((1,1)\). Every map to the singleton is unique. The horizontal and vertical maps out of \(S\), listed in order of its elements, are

Verify that this gives a bifunctor. Compute its limit in both iterated orders, all four projections of the resulting object, and the canonical comparison between the orders.

Solution. Each index has only identities and its two displayed nonidentity arrows. The separate functor laws therefore concern units only. Each mixed square ends at the singleton, so both composites are its unique function. The complete mixed-arrow/currying construction retained in Section 5 then gives a bifunctor on every product arrow.

The horizontal equalizer at vertical object \(0\) is \(S_H=\{0,2,4,5\}\), because these are exactly the columns with \(h_0=h_1\). At vertical object \(1\), the two maps \(B\to1\) agree, so their equalizer is \(B\) with identity inclusion. The induced two maps \(S_H\to B\) are the restrictions of \(v_0,v_1\). Their equalizer is \(\{0,4\}\).

In the other order, the vertical equalizer at horizontal object \(0\) is \(S_V=\{0,1,3,4\}\). At horizontal object \(1\), the equalizer of the two maps \(A\to1\) is \(A\). The induced maps \(S_V\to A\) are the restrictions of \(h_0,h_1\), whose equalizer is again \(\{0,4\}\).

The four projections are inclusion into \(S\), the constant-zero map into \(A\), the map \(0\mapsto0,4\mapsto1\) into \(B\), and the unique map to the singleton. They form a cone by the table. Any set cone has a map \(f:T\to S\) satisfying both equalities \(h_0f=h_1f\) and \(v_0f=v_1f\). Its values lie in \(S_H\cap S_V=\{0,4\}\), giving a unique factor through that inclusion. Its maps to \(A,B,1\) are forced by the cone equations. This proves the full universal property, as well as both iterated computations. Their comparison is the identity on these two elements, since it must preserve the inclusion into \(S\).

Exercise 3 (advanced: a colimit of labelled free spaces)

Let \(k\) be any field. Let \(F:\mathsf{Set}_{\mathcal U}\to k\text{-}\mathsf{Vect}_{\mathcal U}\) take a set \(S\) to the free vector space with basis \([s]\), and a function to the linear map on those basis vectors. Put \(Y=k\). In \((F\downarrow Y)\), take the walking-arrow diagram from \(S=\{a,b\}\) with label map \([a],[b]\mapsto1\), to \(T=\{c\}\) with label map \([c]\mapsto1\), using the function \(a,b\mapsto c\). Determine its colimit and its universal factors. Also determine the colimit of the empty diagram in this comma category.

Solution. A function defines the stated linear map uniquely on basis vectors. Identity and composite functions give identity and composite linear maps by the same uniqueness. The free-space construction has the full adjunction \[ \begin{gathered} \operatorname{Hom}_k(F(S),W)\\ \simeq\operatorname{Hom}_{\mathsf{Set}}(S,UW). \end{gathered} \tag{11.1} \] A linear map restricts to its values at \([s]\); a function extends by finite sums \(\sum_s\lambda_s[s]\mapsto\sum_s\lambda_s f(s)\). Every vector has unique finite coordinates, so the extension is linear, both rules are inverse, and precomposition of functions and postcomposition of linear maps commute with evaluation. This proves the adjunction, including empty \(S\). Theorem 6.1 then proves that \(F\) preserves small colimits.

The stated function gives a comma arrow because each basis label on \(S\) is the composite of its image's label on \(T\). The underlying Set colimit is \(T\) with the displayed function and \(1_T\): a cocone out of a walking arrow is a function out of its target, with the source function forced by composition. Its comma-colimit label is exactly \([c]\mapsto1\), by Theorem 9.1.

More explicitly, a cocone into \((Z,\ell:F(Z)\to k)\) includes a function \(g:T\to Z\). Its source function is \(g(a\mapsto c,b\mapsto c)\), and its comma condition is \(\ell([g(c)])=1\). The unique universal factor from the proposed colimit is that function \(g\); this equation is precisely its comma-arrow condition. Thus no additional factors or constraints have been omitted. For the empty diagram, the underlying initial set is \(\varnothing\), its free space is zero, and its label is the unique zero map to \(k\). There is exactly one function \(\varnothing\to Z\), and its induced zero-space map satisfies the comma condition. Hence this is the initial object and empty colimit.

Exercise 4 (expert: factorization turns a square into a limit)

Let \(I\) be the walking arrow \(s:0\to1\). Let \(\alpha(s):\{a,b\}\to\{c\}\) be its unique function. Let \(\beta(s):\{0,1\}\to\{u,v\}\) send both elements to \(u\). List all objects and arrows of \(\mathsf{Fac}(I)\), compute the presheaf (10.2) on each, and find its limit. Count and list the resulting natural transformations \(\alpha\to\beta\).

Solution. The objects are \(1_0,s,1_1\). There are their three identities and exactly the two arrows \(s\to1_0\), given by \((1_0,s)\), and \(s\to1_1\), given by \((s,1_1)\). To check completeness, an arrow \(f:i\to j\) to \(g:i'\to j'\) requires arrows \(i\to i'\) and \(j'\to j\). For \(f=1_0\), these force \(i'=j'=0\); for \(f=1_1\), they force \(i'=j'=1\). For \(f=s\), the possible target objects are exactly the three listed, with the unique pairs just given. There is no composite of two nonidentity arrows, so these data also display every composition.

The values at \(1_0,s,1_1\) have respectively \(4,4,2\) elements: \[ \begin{gathered} H(1_0)=\{0,1\}^{\{a,b\}},\\ H(s)=\{u,v\}^{\{a,b\}},\\ H(1_1)=\{u,v\}^{\{c\}}. \end{gathered} \tag{11.2} \] The presheaf map \(H(1_0)\to H(s)\) postcomposes with \(\beta(s)\), so sends all four functions to the constant-\(u\) function. The map \(H(1_1)\to H(s)\) precomposes with \(\alpha(s)\), so sends \(c\mapsto u\) to the constant-\(u\) function and \(c\mapsto v\) to the constant-\(v\) function.

A compatible family therefore has an arbitrary function \(\eta_0:\{a,b\}\to\{0,1\}\), the forced function \(\eta_1(c)=u\), and the forced middle value constant \(u\). There are exactly four. List them by the pairs \((\eta_0(a),\eta_0(b))=(0,0),(0,1),(1,0),(1,1)\), with that common \(\eta_1\). Each satisfies the naturality equation \(\beta(s)\eta_0=\eta_1\alpha(s)\), since both sides are constant \(u\). These are precisely the four transformations, and the limit projections are their three displayed component choices. Omitting the reversed second side in (10.1) would give a different diagram and would lose this compatible-family calculation.

12. References and retained proof interfaces

• [Stacks, Adjoints] The Stacks Project authors, Categories, lemma-adjoint-exact, including its complete proof: pinned structured source, lines 3624–3660. Its licence is GFDL 1.2 or later. The linked proof retains that licence; no source expression is incorporated here. The same chapter's set-limit description and omitted-proof statements were comparison context, not additional imported proofs.
• [Yoneda and size] Points, fibres and universal representations, Products, disjoint unions and mixed functors, and Universes and small categories, with the complete Hom-encoding, representation, universe-transfer and connected-constant interfaces specified above. Solution sets and universal representations, Section 1 supplies the full universal-element and element-colimit proofs.
• [Functor and module interfaces] Natural transformations and composition of functors, Passing maps across an adjunction, Hom, tensor and module limits, and Kernels and cokernels, whose complete proved constructions are retained at their named specializations.
• [Structured arrows] Zero maps, components and subobjects, Initial objects and universal colimits, and Building categories from familiar structures, for the exact comma, slice, connected-component and idempotent-shape constructions used above.