Formal colimits and their test objects

Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI (GPT-6.1 Sol, Ultra). Original text public domain (CC0).

Taking a colimit after embedding objects into presheaves changes what counts as a test object. Evaluation at an original object commutes with that colimit. Maps from an arbitrary presheaf need not. This distinction also explains why a represented presheaf colimit gives a universal construction preserved by every functor, while an ordinary colimit often does not.

We use a fixed Grothendieck universe \(\mathcal U\), ambient choice, and a locally \(\mathcal U\)-small category \(C\), whose object set may be larger. All category data are ambient sets. The complete encodings, both Yoneda proofs and element-category laws are retained from Points and representations. The pointwise constructions and all their universal factors are retained from Colimits as connected components. “Small” below means \(\mathcal U\)-small. Enlargements used for ambient transformation sets do not permit larger diagrams to count as small.

1. Put the colimit in the right category

Write

\[ \begin{gathered} \widehat C=\operatorname{Fun} (C^{\mathrm{op}},\mathsf{Set}_{\mathcal U}),\\ C^\vee=\operatorname{Fun} (C,\mathsf{Set}_{\mathcal U})^{\mathrm{op}}. \end{gathered} \tag{1.1} \]

The encoded point functors are \(h_X(T)=\operatorname{Hom}_C(T,X)\) and \(q_X(T)=\operatorname{Hom}_C(X,T)\), with the full Hom encoding understood when literal Hom sets are only small up to bijection. The covariant \(q_X\), regarded in the opposite category, gives the fully faithful functor \(k:C\to C^\vee\); precomposition by \(f:X\to Y\) is its underlying transformation \(q_Y\to q_X\).

For a small \(I\) and \(D:I\to C\), define its formal colimit

\[ \begin{gathered} \mathsf I_C(D)=\operatorname{colim}_{i\in I}h_{D(i)}\\ \text{in }\widehat C. \end{gathered} \tag{1.2} \]

This is also called the quoted ind-lim, even when \(I\) is not filtered. The word does not assert that this presheaf belongs to \(\operatorname{Ind}(C)\). For a diagram \(\alpha:I\to\widehat C\), its quoted colimit instead means the actual colimit of \(\alpha\) itself, without first restricting to original objects.

The complete pointwise presheaf theorem in Colimits as connected components, Section 4, gives

\[ \begin{gathered} \mathsf I_C(D)(T)\\ \simeq \operatorname{colim}_{i\in I} \operatorname{Hom}_C(T,D(i)). \end{gathered} \tag{1.3} \]

In literal Hom notation, a value is represented by a class \([i,u]\), with \(u:T\to D(i)\). A \(C\)-arrow \(a:S\to T\) sends this to \([i,ua]\). An \(I\)-arrow \(s:i\to i'\) supplies the generating identification \([i,u]=[i',D(s)u]\). The retained quotient and Hom proofs verify every functor law, every generator and the full natural cocone factor. A transformation \(t:D\to E\) gives the value map \([i,u]\mapsto[i,t_i u]\), commuting with every \(C\)-arrow by associativity.

For general \(\alpha:I\to\widehat C\), the same pointwise theorem gives \((\operatorname{colim}\alpha)(T)=\operatorname{colim}_i\alpha(i)(T)\), with its exact class maps. Yoneda then supplies

\[ \begin{gathered} \operatorname{Nat}(h_T,\operatorname{colim}\alpha) \simeq\\ \operatorname{colim}_{i\in I} \operatorname{Nat}(h_T,\alpha(i)). \end{gathered} \tag{1.4} \]

It identifies the canonical comparison, not merely the cardinalities: a transformation at stage \(i\), corresponding to \(x\in\alpha(i)(T)\), maps to the transformation corresponding to its class \([i,x]\). Both Yoneda maps are natural in \(T\) and in the diagram, so this equality of maps is natural as well. This guarantee is for every original representable \(h_T\); it is not a classification of all test presheaves that might preserve colimits.

Quoted coproducts are coproducts in \(\widehat C\), with values the tagged disjoint unions of the input values and the full pointwise factor property. A binary quoted disjoint union means this same coproduct. It does not presume a coproduct in \(C\).

If \(L=\operatorname{colim}_I D\) exists in \(C\), with \(c_i:D(i)\to L\), there is a specified comparison

\[ \begin{gathered} \mathsf I_C(D)\longrightarrow h_L,\\ {}[i,u]\longmapsto c_i u. \end{gathered} \tag{1.5} \]

The \(C\)-cocone equations respect each generating relation, and associativity makes this map natural in the test object. It is the unique presheaf-cocone factor of the maps \(h_{c_i}\), by the full universal factor theorem. It need not be invertible.

The formal ordinary-colimit functor instead sends \(T\) to the set of \(C\)-cocones from \(D\) to \(T\), an object of \(C^\vee\). It represents an ordinary colimit when that object exists. This outgoing-cocone object differs from the presheaf in (1.2), whose values are colimits of incoming Hom sets.

2. Hom out of the whole colimit

For any \(\alpha:I\to\widehat C\) and \(B\in\widehat C\), the complete colimit factor property in the presheaf category gives

\[ \begin{gathered} \operatorname{Nat}(\operatorname{colim}_I\alpha,B) \simeq\\ \lim_{i\in I^{\mathrm{op}}} \operatorname{Nat}(\alpha(i),B). \end{gathered} \tag{2.1} \]

The map restricts a transformation along every coprojection. Its inverse is the unique factor of the compatible family. The complete universal cone/cocone proof in Compatible families, Section 2, applied in \(\widehat C\), verifies both inverses and postcomposition naturality in \(B\). For arbitrary \(\alpha,B\), these transformation sets are ambient sets; no smallness of all \(\operatorname{Nat}(A,B)\) is asserted.

For \(D:I\to C\), Yoneda identifies each right-hand term with \(B(D(i))\), with precomposition along \(D(s)\) on the \(I^{\mathrm{op}}\)-arrows. These values and their small compatible-family limit are small. Thus maps from a small colimit of original representables to any \(B\) form a small set, even when general presheaf transformation sets can be larger.

For small \(D:I\to C\) and \(E:J\to C\), combine (2.1) with the representable evaluation (1.4):

\[ \begin{gathered} \operatorname{Nat}(\mathsf I_C(D),\mathsf I_C(E)) \simeq\\ \lim_{i\in I^{\mathrm{op}}} \operatorname{colim}_{j\in J} \operatorname{Hom}_C(D(i),E(j)). \end{gathered} \tag{2.2} \]

The \(I\)-direction is contravariant: \(s:i\to i'\) induces precomposition by \(D(s)\), from the term at \(i'\) to the term at \(i\). The \(J\)-direction is covariant, by postcomposition with \(E(t)\). The full Hom bifunctor and Yoneda proofs make these actions commute, and the full universal factor gives exactly the compatible family of resulting classes. Transforming \(D\) acts by precomposition; transforming \(E\) acts by postcomposition. Naturality of the retained two-variable Yoneda bijections and of the factor map proves both compatibilities.

Each \(i\) may use its own target stage \(j\), and a compatibility is equality of colimit classes. There is no uniform-stage assertion for an arbitrary \(J\). No filteredness or compression of zigzags to one later stage is imposed. Empty indices are included by the existing empty product and empty quotient proofs.

The direction in (1.4) matters. Take the category \(C\) with one object and only its identity. Its presheaf category is \(\mathsf{Set}_{\mathcal U}\), by the complete inverse evaluation construction in Points, Section 8, and its sole representable is \(1\). For the discrete two-object diagram of this representable, the colimit is \(2\). Using the test presheaf \(A=2\), the canonical map is

\[ \begin{gathered} \operatorname{Nat}(2,1)\sqcup \operatorname{Nat}(2,1)\\ \longrightarrow\operatorname{Nat}(2,2). \end{gathered} \tag{2.3} \]

Its domain has two elements, while its codomain has four. The images are the two constant functions, missing the identity and the transposition. Thus arbitrary test presheaves do not have the unconditional evaluation property, even when every diagram value is an original representable.

3. Reverse the whole construction

For a small \(I\) and \(\beta:I^{\mathrm{op}}\to C\), define the formal projective limit, or quoted pro-lim,

\[ \mathsf P_C(\beta)= \lim_{I^{\mathrm{op}}}k\beta \quad\text{in }C^\vee. \tag{3.1} \]

The opposite on \(C^\vee\) reverses the universal arrows. Its underlying covariant functor is the pointwise colimit

\[ \begin{gathered} Q_\beta(T)=\\ \operatorname{colim}_{i\in I} \operatorname{Hom}_C(\beta(i),T). \end{gathered} \tag{3.2} \]

For \(s:i\to i'\) in \(I\), the actual \(C\)-arrow is \(\beta(s):\beta(i')\to\beta(i)\); precomposition gives \(\operatorname{Hom}(\beta(i),T)\to\operatorname{Hom}(\beta(i'),T)\). The class maps in \(T\) are postcomposition. These are the full mixed actions retained from the encoded Hom bifunctor.

Precisely, \(\widehat{C^{\mathrm{op}}}\) is the covariant functor category \(\operatorname{Fun}(C,\mathsf{Set}_{\mathcal U})\). Its identity on objects, followed by passage to the opposite, is a contravariant identification \(\xi:\widehat{C^{\mathrm{op}}}\to C^\vee\). For a transformation \(t:A\to B\), \(\xi(t)\) has direction \(\xi(B)\to\xi(A)\). Its identity law and equation \(\xi(tu)=\xi(u)\xi(t)\) are exactly the definition of opposite composition. Applying the identification twice restores every arrow. Consequently

\[ \begin{gathered} \mathsf P_C(\beta)\simeq\\ \xi\bigl(\mathsf I_{C^{\mathrm{op}}} (\beta^{\mathrm{op}})\bigr). \end{gathered} \tag{3.3} \]

Here \(\beta^{\mathrm{op}}:I\to C^{\mathrm{op}}\). Its Yoneda values are \(\operatorname{Hom}_{C^{\mathrm{op}}}(T,\beta(i)) =\operatorname{Hom}_C(\beta(i),T)\). Thus the underlying value functor is exactly (3.2). Reversing each coprojection gives the specified projection to \(k\beta(i)\). The complete opposite-category factor proof in Compatible families, Section 3, converts every colimit factor and its uniqueness into the projective-limit factor. This proves (3.3) with its structural maps, not merely its object values.

Covariant Yoneda evaluates maps to an original test object:

\[ \begin{gathered} \operatorname{Hom}_{C^\vee} (\mathsf P_C(\beta),kT) \simeq\\ \operatorname{colim}_{i\in I} \operatorname{Hom}_C(\beta(i),T). \end{gathered} \tag{3.4} \]

The canonical comparison sends a map from \(k\beta(i)\) to \(kT\) to its composite with the projection. Under Yoneda this is its class \([i,u]\). For every \(B\in C^\vee\), the whole universal property instead gives

\[ \begin{gathered} \operatorname{Hom}_{C^\vee} (B,\mathsf P_C(\beta)) \simeq\\ \lim_{i\in I^{\mathrm{op}}} \operatorname{Hom}_{C^\vee}(B,k\beta(i)). \end{gathered} \tag{3.5} \]

Its inverse factors the compatible incoming cone, and its forward map composes with every projection. Opposite-category reversal of (2.1) proves both inverse equations and all test-object naturality. The same reasoning applies to a general small \(\beta:I^{\mathrm{op}}\to C^\vee\), where the pro-lim means its actual limit and the terms use \(\beta(i)\) itself.

Quoted products are products in \(C^\vee\). Their underlying covariant value functors are coproducts, with the full opposite universal factor property. The empty quoted projective limit is terminal in \(C^\vee\), represented by its underlying everywhere-empty covariant functor. The empty quoted colimit in \(\widehat C\) is initial, represented by the empty presheaf.

General \(C^\vee\) tests need not satisfy (3.4). For the one-object identity \(C\), \(C^\vee=\mathsf{Set}_{\mathcal U}^{\mathrm{op}}\). The formal projective limit of two copies of the sole original object has underlying set \(2\). With the test object having underlying set \(2\), the canonical map from the coproduct of the two stage Hom sets again has size \(2\to4\), and its images are the two constant functions. This is the entire dual of the preceding counterexample, with Hom reversed.

4. Every presheaf has its own pieces

For \(A\in\widehat C\), let \(E_A\) be its category of elements. An object is \((X,a)\), with \(a\in A(X)\), and an arrow \(u:(X,a)\to(Y,b)\) satisfies \(A(u)b=a\). Retain the full typed identity, composition and size proofs in Points, Section 3. The projection \(j_A:E_A\to C\) sends this arrow to \(u\), and each object supplies its Yoneda transformation \(h_X\to A\).

Theorem 4.1. The whole diagram of these representables has colimit \(A\) in \(\widehat C\):

\[ A\simeq\operatorname{colim}_{(X,a)\in E_A}h_X. \tag{4.1} \]

The index may not be essentially small in \(\mathcal U\). This is a particular large-index colimit whose existence is proved, rather than an assertion that \(\widehat C\) has every large colimit.

Complete canonical proof interface and size bridge. Retain the main density proof of the pinned Stacks lemma-colimit-representable. Its complete main argument uses the category of pairs \((X,a)\) and the pointwise inverse supplied by \((T,t)\) with its identity arrow. Its additional covering-subfamily refinement is not needed here.

Here are all the specialization and universe checks. Choose one larger universe containing the complete data of \(C,E_A\) and \(\mathcal U\). Regard \(C\) as a site with the trivial topology. Every presheaf is then a sheaf, and sheafification is the identity: the singleton identity covering imposes only its own tautological compatibility equation. The retained main proof therefore applies to the actual \(E_A\)-diagram of \(h_X\), without filtering or replacing that category by a small \(\mathcal U\)-subcategory.

Its value comparison and inverse, written in our labels, are

\[ \begin{gathered} {}[(X,a),u:T\to X]\longmapsto A(u)a,\\ t\longmapsto[(T,t),1_T]. \end{gathered} \tag{4.2} \]

The first formula respects every generator because an element arrow \(v:(X,a)\to(Y,b)\) has \(A(v)b=a\), so \(A(vu)b=A(u)a\). The first composite sends \(t\) to itself. For the other composite, \(u\) is the element arrow \((T,A(u)a)\to(X,a)\); its generating identification sends the identity representative to the given \([(X,a),u]\). For \(w:S\to T\), the forward formula on the restricted class is \(A(uw)a=A(w)A(u)a\), proving naturality in every test object. These checks identify the canonical proof's exact inverse and presheaf cocone with our chosen maps.

The resulting values are precisely the given \(A(T)\), which are \(\mathcal U\)-small. Thus the representative can be \(A\) itself. For any \(\mathcal U\)-valued target presheaf \(B\), the canonical universal cocone factor can be taken in the larger presheaf category. The full faithfulness of universe inclusion from Points, Section 7, lifts that factor uniquely to a transformation \(A\to B\) in \(\widehat C\), and lifts each factor equation unchanged. This proves the full colimit property in the original value universe. In particular it does not require the ambient transformation set \(\operatorname{Nat}(A,B)\) to be small in \(\mathcal U\).

Explicitly, for a cocone \(b_{(X,a)}:h_X\to B\), its factor sends \(t\in A(T)\) to \(b_{(T,t),T}(1_T)\). For \(w:S\to T\), the element arrow \((S,A(w)t)\to(T,t)\) and the cocone equation identify \(b_{(S,A(w)t),S}(1_S)\) with \(b_{(T,t),S}(w)\). Naturality of the latter Yoneda transformation makes this \(B(w)b_{(T,t),T}(1_T)\), proving the factor's naturality. For any \(u:T\to X\), the same element-arrow equation makes its value at \(A(u)a\) equal to \(b_{(X,a),T}(u)\), so every cocone leg factors. Conversely the leg at \((T,t)\), evaluated at \(1_T\), forces the factor's value at every \(t\); hence it is unique. Postcomposing \(B\to B'\) applies the same component to each displayed value, proving target naturality and both inverse equations for the cocone-factor bijection.

If \(C\) is empty, \(E_A\) is empty and its presheaf category is the terminal category; its unique object is the empty-index colimit there. If \(C\) is nonempty and \(A\) is the empty presheaf, \(E_A\) is again empty and (4.1) is the initial empty presheaf. No element or representative is invented in these cases. \(\square\)

There is a second complete existing interface for a small \(D:I\to C\). Let \(A=\mathsf I_C(D)\) and let \(d_i\in A(D(i))\) be the class of \(1_{D(i)}\). Its canonical lift is

\[ \begin{gathered} \widetilde D:I\to E_A,\\ i\longmapsto(D(i),d_i). \end{gathered} \tag{4.3} \]

It sends \(s\) to \(D(s)\). The generating quotient equation says \(A(D(s))d_{i'}=d_i\), so that is an element arrow; identities and composites are exactly those of \(D\).

Retain the entire proof of Ind-objects through their elements, Proposition 1.2. Its statement allows every small \(I\), without filteredness. An incoming comma at \((T,t)\) is precisely the category of representatives \(u:T\to D(i)\) of \(t\). The existing proof obtains nonemptiness from the quotient representatives and connectedness from every generated finite zigzag, all intermediate representatives still having class \(t\). Therefore \(\widetilde D\) is cofinal. Its empty-index argument makes the same assertion vacuous when \(E_A\) is empty.

This proof does not require \(E_A\) to be small. The full arbitrary-ambient cocone theorem in Change an index, Section 2 applies to this lift with its actual large target index. Nor does it make \(A\) an ind-object: that additionally requires a small filtered presentation.

5. A represented formal colimit is absolute

Theorem 5.1. Let \(D:I\to C\) be small, and suppose a specified isomorphism \(\rho:\mathsf I_C(D)\simeq h_X\) exists. Transport the coprojections through \(\rho\); full faithfulness of Yoneda gives their \(C\)-cocone \(c_i:D(i)\to X\). For every functor \(F:C\to C'\) between locally small categories, the cocone \(F(c_i)\) makes \(F(X)\) a colimit of \(FD\).

Exact complete owned proof interface. Retain Ind-objects through their elements, Proposition 3.2 in full. It allows an arbitrary small diagram and an arbitrary functor \(F\), with no filteredness or general existence of colimits in \(C'\). Its test presheaf is \(H_Y(T)=\operatorname{Hom}_{C'}(F(T),Y)\).

The target Hom sets need not be small in the original \(\mathcal U\). Choose one larger universe containing the entire category data of \(C,C'\), the diagram and \(\mathcal U\), using the enlargement axiom in Universes, Section 1. The independently proved preservation theorem in Section 6 below carries the specified \(\mathcal U\)-small formal colimit and its representation to this larger value universe. Now \(H_Y\) is a presheaf in that universe, so the retained proof applies to it there. With this size change understood, its bijections are

\[ \begin{gathered} \operatorname{Hom}_{C'}(F(X),Y) \\\simeq\operatorname{Nat}(h_X,H_Y)\\ \simeq\operatorname{Nat}(\mathsf I_C(D),H_Y)\\ \simeq \lim_{i\in I^{\mathrm{op}}} \operatorname{Hom}_{C'}(F(D(i)),Y). \end{gathered} \tag{5.1} \]

The existing proof explicitly identifies this map with composition by every \(F(c_i)\), naturally in \(Y\); its bijectivity is the full factor and uniqueness property. For arbitrary Hom labels use the complete encoded Hom and Yoneda proofs in Points, Sections 1–2: \(H_Y(v)\) decodes a map, precomposes by \(F(v)\), then encodes. Cancellation of the codes gives every contravariant functor equation and target-postcomposition naturality. Thus these same complete bijections apply to locally small categories with arbitrary labels. \(\square\)

Taking \(F=1_C\) shows that \(X\) is the ordinary colimit too. The converse is false. In the one-object identity \(C\), the discrete two-copy diagram has the ordinary colimit equal to its sole object: every ordinary cocone and every factor is unique. Its formal colimit is the two-element presheaf, while \(h_X=1\). The functor \(F:C\to\mathsf{Set}\) selecting \(2\) sends both ordinary coprojections to \(1_2\). They are not a coproduct cocone in sets: the cocone with first leg \(1_2\) and second leg the constant-zero function has no common factor through those two identities.

For an empty \(I\), the formal colimit is the empty presheaf. It cannot be represented by any \(X\in C\), since \(h_X(X)\) contains \(1_X\). Thus the representation premise of Theorem 5.1 is impossible in that case; the theorem does not assert that every functor preserves initial objects.

6. Change the universe with the values fixed

Let \(\mathcal U\subseteq\mathcal V\) and keep \(C\) fixed and locally \(\mathcal U\)-small. The inclusion \(j:\widehat C_{\mathcal U}\to\widehat C_{\mathcal V}\) is fully faithful, by the complete component and postcomposition proof in Points, Section 7. This remains true when the common transformation sets are larger than \(\mathcal U\).

Proposition 6.1. The inclusion \(j\) preserves every limit and colimit indexed by a \(\mathcal U\)-small category, with their specified structural maps.

Proof. Use the complete arbitrary-base pointwise theorem in Colimits as connected components, Section 4. At \(T\), a small colimit is the quotient of the small tagged union by all diagram-arrow identifications. Retain its chosen \(\mathcal U\)-encoding and regard it as a \(\mathcal V\)-set. The actual relation, classes and coprojections are unchanged. The entire quotient factor proof applies to every target set, including a \(\mathcal V\)-set: each compatible family descends to the same unique class map. The full pointwise presheaf naturality proof then supplies the unique transformation to any \(\mathcal V\)-valued target presheaf.

Likewise a small limit is the set of compatible tuples in the small product of its values. The same tuples and projections form a \(\mathcal V\)-set. Every incoming cone, even from a \(\mathcal V\)-set, has the same unique tuple-valued factor. The full pointwise functor-category proof verifies all naturality equations, so these are the chosen presheaf limits in the larger universe too. Both comparisons respect every original projection or coprojection. This proves preservation with the universal maps, including empty diagrams, without requiring a small object set for \(C\). \(\square\)

This assertion retains the \(\mathcal U\)-bound on the index. For the one-object identity \(C\), the discrete family of singleton presheaves indexed by the ambient set \(\mathcal U\) is small in a universe containing \(\mathcal U\). Its larger-universe coproduct has value \(\mathcal U\), which is not \(\mathcal U\)-small by the complete Cantor proof in Universes, Lemma 1.1. It therefore cannot be the image of a smaller-valued presheaf. The exact Set quotient-size criterion in Colimits as connected components, Section 6, also proves that no such coproduct exists in the smaller presheaf category.

7. Four graded exercises with full solutions

Exercise 1 — the four maps

In the one-object identity category, take the discrete two-copy diagram of the original object. For the test presheaf \(2\), identify every map to its formal colimit and identify the image of the stagewise comparison. Repeat for the formal projective limit in \(C^\vee\).

Solution. The presheaf category is sets, its original object is \(1\), and the formal colimit is \(1\sqcup1=2\). A function \(2\to2\) is given by its ordered pair of values, so the four functions have pairs \((0,0),(0,1),(1,0),(1,1)\). Each stage has one function \(2\to1\). Postcomposing with its coprojection gives respectively \((0,0)\) and \((1,1)\). The identity and transposition, with pairs \((0,1),(1,0)\), are missing.

In \(C^\vee=\mathsf{Set}^{\mathrm{op}}\), the formal projective limit of the two original values has underlying set \(2\). Its maps to the test object with underlying set \(2\) are the same four functions from the test's underlying set to the limit's underlying set. Composing a stage map with a projection reverses to postcomposing \(2\to1\) with the corresponding Set coprojection. Hence the two constant functions are again the entire image. This checks the dual Hom direction as well as the cardinalities.

Exercise 2 — two elements sharing one restriction

Let \(C=(x\to y)\), with arrow \(a:x\to y\). Let \(A(y)=\{r,s\}\), \(A(x)=\{u\}\), with both elements restricting to \(u\). Describe \(E_A\), compute the entire density diagram pointwise, and give the factor of every presheaf cocone to \(B\).

Solution. The objects of \(E_A\) are \((x,u),(y,r),(y,s)\). In addition to their identities, its arrows are the two copies of \(a\) from \((x,u)\) to each of the other objects. There is no arrow between the latter two: the only \(C\)-endomorphism of \(y\) is its identity, and it cannot carry one distinct specified element to the other.

The density diagram is \(h_x\to h_y\) along each branch. At \(x\) it is \(1\to1\) and \(1\to1\), so its generated quotient has one class. At \(y\) it is \(\varnothing\to1\) and \(\varnothing\to1\), so its quotient has two classes. The restriction from \(y\) to \(x\) sends both to the single class. These are exactly \(A\) and its restriction.

By full Yoneda, a cocone to \(B\) is a choice \(b,c\in B(y)\) and \(d\in B(x)\) with \(B(a)b=d=B(a)c\). Its factor \(A\to B\) sends \(r\) to \(b\), \(s\) to \(c\), and \(u\) to \(d\). The stated equalities are precisely its naturality at \(a\). All components are forced, so this factor is unique; postcomposing \(B\to B'\) applies its component functions to \(b,c,d\), proving target naturality. This also handles empty values: if such choices do not exist, neither a cocone nor a factor exists.

Exercise 3 — one target stage for each source piece

Again use the one-object identity \(C\). Let \(D\) have discrete two-object index and \(E\) discrete three-object index, all values equal to the original object. Compute (2.2), compare it with first taking the limit over the two source indices, and include the empty-index boundaries.

Solution. The two formal colimits are the sets \(2\) and \(3\), so their transformation set has \(3^2=9\) elements. Each original Hom set is a singleton. Taking its colimit over the three target indices gives \(3\), and taking the limit over the discrete two source indices gives \(3\times3\), again nine elements. The bijection sends a function to its separate values at the two source elements.

If one first takes the limit over the two source indices at each fixed target index, the result is a singleton at each of the three target indices. Its colimit is \(3\). Its canonical map into \(3\times3\) sends \(j\) to \((j,j)\), missing all six pairs with different coordinates. Requiring one uniform target stage therefore changes the answer.

If the source index is empty, its formal colimit is the empty set, and maps from it form a singleton. The right side is the empty compatible-family product, also a singleton. If the target index is empty and the source has two objects, there are no maps \(2\to\varnothing\), and the right side is the product of two empty sets. If both indices are empty, there is one empty function, agreeing with the empty product.

Exercise 4 — split an idempotent before applying any functor

Suppose \(e:X\to X\) satisfies \(e^2=e\), and is split by maps \(r:X\to R\), \(i:R\to X\), with \(ri=1_R\) and \(ir=e\). Use the one-object index category whose endomorphisms are \(1,e\). Prove its \(C\)-diagram has formal colimit \(h_R\), and verify directly that every functor preserves the resulting colimit.

Solution. At a test \(T\), the formal colimit is the quotient of \(\operatorname{Hom}_C(T,X)\) by the generated identifications \(f\sim ef\). Send \([f]\) to \(rf:T\to R\). This respects every generator because \(re=rir=r\). Its inverse sends \(g:T\to R\) to \([ig]\). The first composite is \(rig=g\); the second is \([irf]=[ef]=[f]\). For \(v:S\to T\), both routes send \([f]\) to \(rfv\), so the bijection is natural. It identifies the formal coprojection with \(h_r:h_X\to h_R\), hence supplies the specified formal representation.

For any functor \(F\), write \(E=F(e)\), \(P=F(i)\), \(Q=F(r)\). Then \(QP=1\) and \(PQ=E\). A cocone on the one-object idempotent diagram is exactly \(b:F(X)\to Y\) with \(bE=b\). Its factor through \(Q\) is \(bP:F(R)\to Y\), since \(bPQ=bE=b\). If \(zQ=b\), then \(z=zQP=bP\), proving uniqueness. These factors are natural under postcomposition in \(Y\). Thus \(F(R)\), with \(F(r)\), is the actual colimit, without any hypothesis that \(F\) preserves other colimits.

8. References and retained proof interfaces