Change an index without changing its universal maps

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).

One can often compute a colimit using fewer stages. Reaching every stage is only part of the requirement: different ways of reaching it must also impose the same map to the proposed target. Connected comma categories express exactly this second condition. They let us extend every cocone, with all its maps, rather than merely compare two candidate objects.

We use the complete comma and component constructions in Zero maps, components and subobjects, the quotient and representable calculations in Colimits as connected components, and the universal cone and cocone functors in Compatible families and universal cones. A fixed Grothendieck universe \(\mathcal U\), ambient choice and the size conventions of Universes and small categories are understood. All category data are sets in the ambient setting. We specify whenever an object set or a Hom set must be \(\mathcal U\)-small.

1. Connected comma categories

Let \(\phi:J\to I\). For \(i\in I\), the category \((i\downarrow\phi)\) has objects \((j,u)\), where \(u:i\to\phi(j)\). A morphism \(s:(j,u)\to(j',u')\) is \(s:j\to j'\) satisfying

Retain the complete typed identity and composition proof for incoming comma categories in Zero maps, Section 1, with the constant functor at \(i\) on its left. Its identities are those of \(J\), and the composite of \(s\) and \(t\) satisfies the equation because \(\phi(ts)u=\phi(t)\phi(s)u=\phi(t)u'=u''\).

A category is connected when it is nonempty and all its objects are joined by a finite zigzag of arrows, permitting either direction. The full equivalence-relation and opposite-invariance proofs are retained from Zero maps, Section 4. In particular connectedness does not require a common receiver or filteredness.

We call \(\phi\) cofinal when \((i\downarrow\phi)\) is connected for every \(i\in I\). The alternative name is a final functor. We call it co-initial, or co-cofinal, when \(\phi^{\mathrm{op}}\) is cofinal. The opposite of \((i\downarrow\phi^{\mathrm{op}})\) is \((\phi\downarrow i)\): its objects are arrows \(\phi(j)\to i\), and its arrows satisfy the outgoing comma equation. Thus co-initiality means that these outgoing comma categories are connected.

These definitions use the actual ambient comma categories. They impose no working-universe bound on their entire sets of objects.

2. Extend every cocone

Theorem 2.1. If \(\phi:J\to I\) is cofinal and \(D:I\to C\), restriction is a bijection

It is natural in \(T\), and compatible with every transformation of \(D\). The assertion holds for arbitrary ambient index and value categories. Neither colimit is presumed to exist.

Proof. Let \(b_j:D(\phi(j))\to T\) be a cocone on the restriction. For each \(i\), take an object \((j,u)\) of the nonempty comma category and set

This is independent of the choice. For a comma arrow \(s:(j,u)\to(j',u')\), the restricted cocone equation and (1.1) give

The same equality can be read in reverse. Applying it along a finite zigzag identifies the two expressions for any pair of comma objects. No common upper object is used.

For \(v:i\to i'\), choose \((j,u')\in(i'\downarrow\phi)\). Then \((j,u'v)\in(i\downarrow\phi)\), so independence of choices gives

Thus \(e\) is an \(I\)-cocone. At \(i=\phi(j)\), use the comma object \((j,1_{\phi(j)})\); it gives \(e_{\phi(j)}=b_j\). Hence \(R_T(e)=b\).

Conversely, start with an \(I\)-cocone \(a\), restrict it, then apply (2.2). Its equation at \(u:i\to\phi(j)\) gives \(a_{\phi(j)}D(u)=a_i\), so extension recovers \(a\). These are both inverse equations.

For \(w:T\to T'\), extending the cocone \(w b_j\) gives \(w b_jD(u)=w e_i\); this proves naturality in the target. For a transformation \(t:D\to E\) and a restricted cocone \(b:E\phi\to T\), the two routes of the diagram-variable square at \(i\) agree:

Restriction itself has the same compatibility by its component formula. Thus the inverse bijections are natural in both variables. \(\square\)

Corollary 2.2. Either of \(\operatorname{colim}_J D\phi\) and \(\operatorname{colim}_I D\) exists if and only if the other does. If both are chosen, the canonical arrow

is invertible and respects the specified coprojections.

Proof. Suppose \(K\), with coprojections \(c_j\), is the restricted colimit. Extend \(c\) by Theorem 2.1 to \(e_i:D(i)\to K\). For an \(I\)-cocone \(a\) to \(T\), its restriction factors through a unique \(f:K\to T\). Equation (2.2) gives \(f e_i=f c_jD(u)=a_{\phi(j)}D(u)=a_i\). If another arrow factors \(a\), evaluating at \(\phi(j)\), where \(e_{\phi(j)}=c_j\), makes it the same restricted factor. This proves the full \(I\)-colimit property. If the \(I\)-colimit exists first, its restriction represents the restricted cocone functor by (2.1), giving the other existence implication.

Write \(L=\operatorname{colim}_I D\), with maps \(a_i\). The canonical \(\lambda:K\to L\) has \(\lambda c_j=a_{\phi(j)}\). The extended cocone induces \(\mu:L\to K\), with \(\mu a_i=e_i\). Then \(\lambda\mu a_i=\lambda e_i=a_i\) and \(\mu\lambda c_j=e_{\phi(j)}=c_j\). Both universal uniqueness properties prove the inverse identities. Naturality in a transformation of \(D\) follows from (2.5) and these coprojection equations. \(\square\)

For \(\beta:I^{\mathrm{op}}\to C\), apply the entire proof to \(\beta^{\mathrm{op}}:I\to C^{\mathrm{op}}\). The complete cone–cocone reversal in Compatible families, Section 3, identifies its factors and uniqueness with incoming cones in \(C\). Thus, for cofinal \(\phi\),

is the canonical isomorphism whenever either limit exists. It sends a cone to its restricted cone. For co-initial \(\phi\), the dual statement applies to covariant diagrams \(D:I\to C\), giving \(\lim_I D\to\lim_J D\phi\).

3. Six tests for the same property

Assume here that \(I,J\) are \(\mathcal U\)-small. Value categories in the formal-functor statements are locally \(\mathcal U\)-small; their object sets may be larger. The complete Hom encoding and small compatible-family bounds from Compatible families, Sections 2–3, supply the actual universe-member versions of all value functors below.

For \(D\), let \(P_D(T)\) be its set of incoming cones and let \(Q_D(T)\) be its set of outgoing cocones. Retain their full functor and universal-factor proofs from that lesson: \(P_D\) is an object of \(\widehat C=\operatorname{Fun}(C^{\mathrm{op}},\mathsf{Set}_{\mathcal U})\); \(Q_D\) is covariant as a value functor, and is regarded as an object of \(C^\vee=\operatorname{Fun}(C,\mathsf{Set}_{\mathcal U})^{\mathrm{op}}\). These objects exist even if \(C\) supplies no limit or colimit representative.

Theorem 3.1. The following conditions are equivalent.

1. \(\phi\) is cofinal.
2. For every \(\beta:I^{\mathrm{op}}\to\mathsf{Set}_{\mathcal U}\), restriction of compatible tuples gives a bijection \(\lim_{I^{\mathrm{op}}}\beta\to\lim_{J^{\mathrm{op}}}\beta\phi^{\mathrm{op}}\).
3. For every such projective diagram \(\beta\) into every locally \(\mathcal U\)-small \(C\), the canonical transformation \(P_\beta\to P_{\beta\phi^{\mathrm{op}}}\) is an isomorphism in \(\widehat C\).
4. For every \(\alpha:I\to\mathsf{Set}_{\mathcal U}\), the canonical map \(\operatorname{colim}_J\alpha\phi\to\operatorname{colim}_I\alpha\) is bijective.
5. For every \(\alpha:I\to C\) as above, the canonical arrow \(Q_{\alpha\phi}\to Q_\alpha\) is an isomorphism in \(C^\vee\).
6. For every \(i\in I\), the colimit \(\operatorname{colim}_{j\in J}\operatorname{Hom}_I(i,\phi(j))\) is a singleton.

In condition 5 the underlying covariant transformation is restriction \(Q_\alpha\to Q_{\alpha\phi}\). The opposite on \(C^\vee\) reverses that direction.

Proof. Condition 1 implies 4 by Corollary 2.2, since small Set colimits exist. It implies 2 by (2.7). The same cocone bijection, natural in \(T\), is an isomorphism of covariant value functors, hence gives 5 in their opposite category. Applying the full opposite-category proof gives a cone bijection natural in \(T\), hence 3.

To compare 4 and 6, fix \(i\) and set \(H_i(k)=\operatorname{Hom}_I(i,k)\), with postcomposition as its arrow action. The complete covariant Hom proof and identity-class calculation in Colimits as connected components, Section 3, give

Applying 4 to \(H_i\) gives 6. Conversely the covariant element category of \(H_i\phi\) is exactly \((i\downarrow\phi)\): objects are \((j,u:i\to\phi(j))\), and its arrow equation is (1.1). The entire class–component bijection in that lesson, Section 2, identifies its colimit with \(\pi_0(i\downarrow\phi)\), on every generating arrow. Condition 6 says this component set has one element; the retained connectedness convention says precisely that the comma category is nonempty and connected. Thus 6 implies 1.

Condition 2 also implies 6, without presuming a colimit-preservation result. Let \(2=\{0,1\}\) and consider the contravariant diagram

The functor laws follow from the covariant laws of \(H_i\), and these function sets are small by the full universe function-set proof. A compatible tuple in \(\lim\beta_i\) is exactly a Set cocone from \(H_i\) to \(2\). The quotient universal factors in Colimits as connected components, Section 1, therefore identify the restriction map in 2 with

where the map sends a value to the constant function on \(Q_i\). If \(Q_i\) is empty, its two possible values give the same empty function, so this map is not injective. If \(Q_i\) has distinct \(x,y\), the function equal to \(1\) at \(x\) and \(0\) elsewhere is not constant, so the map is not surjective. Bijectivity forces \(Q_i\) to have exactly one element. Hence 2 implies 6.

For 3 implies 2, take \(C=\mathsf{Set}_{\mathcal U}\) and evaluate the presheaf transformation at the singleton. The natural bijection \(\mathsf{Set}(1,S)\simeq S\), sending a function to its value, identifies its component with exactly the compatible-tuple restriction in 2. Its inverse sends \(s\) to the function with that value; both maps commute with every transition.

For 5 implies 4, again take \(C=\mathsf{Set}_{\mathcal U}\). The existing Set colimits represent \(Q_\alpha\) and \(Q_{\alpha\phi}\) by their specified cocone factors. The underlying transformation in 5 is precomposition by the actual canonical arrow \(\lambda\) of condition 4. The full covariant Yoneda and full-faithfulness proof in Points and representations, Section 2 reflects its invertibility, so \(\lambda\) is invertible. Thus 5 implies 4.

We have proved \(1\) implies \(2,3,4,5\), and \(3\Rightarrow2\Rightarrow6\Rightarrow1\) and \(5\Rightarrow4\Rightarrow6\Rightarrow1\). This proves all six equivalences, with the specified comparison maps. \(\square\)

If \(I\) is empty, the existence of \(\phi\) forces \(J\) empty. The comma condition is vacuous and every comparison is the identity on the empty-index universal family. If \(J\) is empty and \(I\) has an object \(i\), \(Q_i\) is empty and the two-point test fails; the unique functor is not cofinal. The same statements, after opposites, describe co-initiality.

4. Compose and cancel index changes

Proposition 4.1. Cofinality has the following consequences.

• It induces a bijection \(\pi_0(J)\to\pi_0(I)\). In particular \(J\) is connected if and only if \(I\) is connected.
• If \(\psi:K\to J\) and \(\phi:J\to I\) are cofinal, then \(\phi\psi\) is cofinal.
• If \(\psi\) and \(\phi\psi\) are cofinal, then \(\phi\) is cofinal.
• If \(\phi\) is fully faithful and \(\phi\psi\) is cofinal, then both \(\psi\) and \(\phi\) are cofinal.

Proof. For small indices, apply Theorem 3.1 to the constant singleton diagram. The full constant-diagram class bijection in Colimits as connected components, Section 3, identifies its canonical colimit map with \([j]\mapsto[\phi(j)]\). Hence the component sets are bijective. A connected category has a singleton component set, while an empty category has an empty component set, giving the first conclusion with its nonemptiness condition.

Write \(a_\phi(\alpha)\) for the canonical Set-colimit map for \(\alpha:I\to\mathsf{Set}\). On the quotient's every representative, \([k,x]\) maps first to \([\psi(k),x]\) and then to \([\phi\psi(k),x]\). Thus the entire canonical maps satisfy

If both factors are bijective for every \(\alpha\), the composite is bijective, so Theorem 3.1 gives the second conclusion. If the composite and the second factor are bijective, then \(a_\phi=a_{\phi\psi}a_\psi^{-1}\) is bijective for every \(\alpha\), giving the third conclusion.

For the fourth conclusion, full faithfulness gives an isomorphism of entire comma categories

Send \((k,v:j\to\psi(k))\) to \((k,\phi(v))\) and leave its underlying \(K\)-arrow unchanged. Conversely every arrow \(\phi(j)\to\phi\psi(k)\) lifts to a unique \(v\), by fullness and faithfulness. A \(K\)-arrow \(t:k\to k'\) satisfies \(\psi(t)v=v'\) if and only if its image satisfies \(\phi\psi(t)\phi(v)=\phi(v')\), by faithfulness. This verifies the inverse on every Hom set. Both functors preserve identities and composites because they leave the underlying arrows unchanged. Cofinality of the composite makes the right comma connected, so \(\psi\) is cofinal. The third conclusion now gives \(\phi\) cofinal.

For arbitrary ambient \(I,J,K\), take one larger universe containing their complete object and arrow data, with their encoding if needed. All three categories are small there, so the arguments just given apply to Set values in that universe. Their conclusions concern only actual comma objects, arrows and finite zigzags, and hence are unchanged by enlargement. This proves the full ambient assertions. It does not assert that an arbitrary large-index colimit has values small in the original \(\mathcal U\). \(\square\)

Applying all four conclusions to opposite categories gives the corresponding statements for co-initial functors.

5. Replace a large index by a small full one

An ambient category \(I\) is cofinally \(\mathcal U\)-small when there is a \(\mathcal U\)-small \(K\) and a cofinal functor \(\theta:K\to I\). It is co-cofinally \(\mathcal U\)-small when \(I^{\mathrm{op}}\) is cofinally \(\mathcal U\)-small. The empty category has both properties, witnessed by its identity. An empty category cannot witness cofinality into a nonempty one, because its incoming commas there are empty.

Theorem 5.1. Let \(I\) be locally \(\mathcal U\)-small and cofinally \(\mathcal U\)-small. Then it has a full \(\mathcal U\)-small subcategory whose inclusion is cofinal.

Proof. Choose a small witness \(\theta:K\to I\), and let \(S\) be the image of its object map. It is \(\mathcal U\)-small: it is bijective to the quotient of \(\operatorname{Ob}(K)\) by “has the same image,” using the full transported-quotient proof in Universes, Section 3. Let \(J\) be the full subcategory on \(S\). It is locally small because its Hom sets are the corresponding Hom sets of \(I\). Its entire arrow set is the tagged union of these small Hom sets over the small set \(S^2\); the same universe proof bounds this union. Thus \(J\), including all its morphisms, is small.

Factor \(\theta\) as

The inclusion \(\phi\) is fully faithful, and \(\phi\psi=\theta\) is cofinal. Proposition 4.1 gives cofinality of both \(\psi\) and \(\phi\). \(\square\)

The locally small hypothesis is the category convention used in the working-universe full-image assertion. It bounds every Hom set, rather than merely the number of chosen objects. For a co-cofinally small locally small \(I\), apply the full proof to \(I^{\mathrm{op}}\) and take opposites. This gives a small full co-initial subcategory of \(I\).

Corollary 5.2. If \(C\) admits all \(\mathcal U\)-small colimits, then it admits every colimit indexed by a cofinally \(\mathcal U\)-small \(I\). If \(C\) admits all small limits, it admits every limit indexed by a co-cofinally small \(I\).

Proof. A small witness \(\theta:K\to I\) suffices; local smallness of \(I\) is unnecessary for this conclusion. For \(D:I\to C\), form the small colimit of \(D\theta\). Theorem 2.1 extends its cocone to every \(i\), and the complete existence and factor argument in Corollary 2.2 makes it the \(I\)-colimit. The cone–cocone reversal gives the limit statement.

Different witnesses and chosen small colimits yield the unique comparison commuting with every extended \(I\)-coprojection. To see its inverse, factor each universal cocone through the other; their composites fix every coprojection, so uniqueness makes them identities. For three choices, the composite comparison and the direct comparison agree on every coprojection, proving the cocycle identity. For the empty index these statements reduce to the full initial-object uniqueness proof; their duals reduce to terminal-object uniqueness. \(\square\)

Without local \(\mathcal U\)-smallness, the full-image conclusion can fail even though small witnesses still compute colimits. For a concrete boundary, let \(M=\{1\}\sqcup\mathcal U\), with identity \(1\), and let the product of any two elements different from \(1\) be a fixed \(0\) in the second summand. This is associative: after removing identity factors, a product of at least two nonidentity factors is \(0\), independently of parentheses. In particular \(0m=m0=0\).

Let \(I\) be its one-object monoid category, and let \(K\) be the one-object submonoid category \(\{1,0\}\). The inclusion \(K\to I\) is cofinal. Its comma objects are \(m\in M\), and the \(K\)-arrow \(0\) sends every \(m\) to \(0m=0\); hence the comma is nonempty and connected. But a full subcategory of \(I\) is either empty or all of \(I\). The empty one is not cofinal, and the other has the nonsmall Hom set \(M\). Indeed \(\mathcal U\) is not \(\mathcal U\)-small by the complete Cantor proof in Universes, Lemma 1.1, and subsets of small sets would be small. Thus neither is a small full cofinal subcategory. This example falls outside the locally small convention in Theorem 5.1.

6. Four graded exercises with full solutions

Exercise 1 — reaching objects does not connect choices

Let \(J\) be discrete with two objects, let \(I\) have one object and only its identity, and let \(\phi:J\to I\) be the unique functor. Every object of \(I\) maps to an image object. Determine cofinality, and exhibit failures of both the Set-colimit and projective-limit tests.

Solution. The sole incoming comma is the discrete two-object category: there is one arrow to each image, but no arrow in \(J\) connecting them. It is not connected, so \(\phi\) is not cofinal.

For the singleton diagram on \(I\), its colimit is a singleton. The restricted diagram on \(J\) has a tagged two-element colimit. The canonical map sends both elements to the singleton, so is not injective.

For the contravariant diagram with value \(2=\{0,1\}\) on \(I\), the limit is \(2\). On \(J\) the limit is \(2\times2\), and restriction is \(x\mapsto(x,x)\). The compatible pair \((0,1)\) is outside its image. Nonempty comma categories alone do not ensure either comparison.

Exercise 2 — choose the endpoint with the correct direction

Let \(I=(0\to1)\). Prove that the inclusion of \(1\) is cofinal and the inclusion of \(0\) is co-initial. Compute the resulting universal maps for a diagram \(A\xrightarrow{f}B\) and for a projective diagram \(B'\xrightarrow{g}A'\) indexed by \(I^{\mathrm{op}}\). Show that the two endpoint inclusions cannot be interchanged.

Solution. For the inclusion of \(1\), each incoming comma contains exactly the unique arrow \(i\to1\), and only its identity. Hence it is cofinal. Its colimit calculation says \(\operatorname{colim}(A\to B)=B\), with coprojections \(f\) and \(1_B\). A cocone \((a:A\to T,b:B\to T)\) has \(a=bf\), so its unique factor is \(b\).

For the inclusion of \(0\), every outgoing comma contains exactly the unique arrow \(0\to i\). Hence it is co-initial. Its limit calculation says \(\lim(A\to B)=A\), with projections \(1_A,f\). A cone \((a:T\to A,b:T\to B)\) has \(b=fa\), so its unique factor is \(a\).

The projective diagram has its actual arrow from value \(B'\) at \(1\) to value \(A'\) at \(0\). Cofinality of the inclusion of \(1\) gives its limit \(B'\), with projections \(1_{B'},g\). Its compatible incoming cones factor uniquely through their component at \(1\).

The inclusion of \(0\) is not cofinal: the incoming comma at \(1\) is empty. The inclusion of \(1\) is not co-initial: the outgoing comma at \(0\) is empty. For the concrete covariant Set diagram \(\varnothing\to1\), choosing \(0\) for its colimit would give \(\varnothing\), while its true colimit is \(1\); choosing \(1\) for its limit would give \(1\), while its true limit is \(\varnothing\).

Exercise 3 — why one cancellation needs full faithfulness

Let \(J=(0\to1)\), let \(I\) be the one-object identity category, and let \(\phi:J\to I\) be its collapse. Let \(K\) be the one-object identity category and let \(\psi:K\to J\) select \(0\). Determine cofinality of \(\phi,\psi,\phi\psi\) and the failed hypothesis of fully faithful cancellation.

Solution. The incoming comma for \(\phi\) is \(J\) itself: there is a unique arrow in \(I\) from its object to each \(\phi(j)\), and all \(J\)-arrows satisfy its equation. It is nonempty and connected, so \(\phi\) is cofinal. The composite is the identity functor between one-object identity categories, so is cofinal.

The incoming comma \((1\downarrow\psi)\) is empty, since there is no arrow \(1\to0\) in \(J\). Thus \(\psi\) is not cofinal. The Hom map \(\operatorname{Hom}_J(1,0)=\varnothing\to\operatorname{Hom}_I(*,*)=1\) is not surjective, so \(\phi\) is not full. This exhibits why cofinality of \(\phi\) and \(\phi\psi\) does not by itself permit cancellation to obtain cofinality of \(\psi\).

Exercise 4 — omit every odd stage

Consider the ordered category \(\mathbb N\), with a unique arrow \(n\to m\) when \(n\le m\). Let \(E\) be its full subcategory on the even numbers. Prove the inclusion is cofinal. Compute its effect on the colimit of \(D(n)=\{0,\ldots,n\}\), with inclusions, and on the limit of the projective diagram of binary strings of length \(n\), with truncation maps.

Solution. At \(n\), the incoming comma is the ordered set of even numbers \(e\ge n\). It is nonempty, since some even number is at least \(n\). Two objects \(e,e'\) have the even common receiver \(\max(e,e')\), so the comma is connected. This verifies cofinality on every \(n\). The full subcategory is small because its objects and unique-arrow sets are countable.

For \(D\), send a quotient class \([n,x]\) to \(x\in\mathbb N\). Transitions preserve \(x\), so this is well-defined. Every \(x\) occurs at stage \(x\). If \([n,x]\) and \([m,y]\) have the same image, \(x=y\), and both representatives map to that same element in stage \(\max(n,m)\), so the map is injective. Thus the colimit is \(\mathbb N\). Over the even stages the same proof uses an even stage at least \(\max(n,m)\), giving the same bijection. The canonical comparison sends each class to its unchanged integer, so is the identity under these bijections.

A cocone to a set \(T\) is a family of functions \(b_n:\{0,\ldots,n\}\to T\) agreeing under inclusions. Its unique factor \(\mathbb N\to T\) sends \(x\) to \(b_n(x)\) for any \(n\ge x\). Independence and uniqueness follow from a common later stage. Restricting to even \(n\) supplies the same formula. This also includes \(T=\varnothing\): neither family nor a factor exists.

Write the binary-string diagram as \(\beta(n)=2^{\{0,\ldots,n-1\}}\), with \(\beta(0)=1\); arrows in \(\mathbb N^{\mathrm{op}}\) truncate a longer string to a shorter one. A compatible family \((s_n)\) determines a unique infinite string \(s:\mathbb N\to2\): for \(r\), choose any \(n>r\) and set \(s(r)=s_n(r)\). Truncation compatibility makes this independent of \(n\). Conversely \(s\) supplies all its finite initial segments; both constructions are inverse.

The even-length family determines the same infinite string by choosing an even \(e>r\). Its missing odd segment of length \(n\) is the truncation of any even segment of length \(e\ge n\), independently of \(e\). Hence restriction to even stages is bijective on compatible families. For an incoming cone from \(T\), apply these constructions separately to each \(t\in T\); the unique factor \(T\to2^{\mathbb N}\) is its resulting infinite string. This proves the full limit property and the exact canonical restriction comparison, including empty \(T\).

7. References and retained proof interfaces

• Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Section 2.6, for cofinal functors and directed colimits.
• The Stacks Project, cofinal and initial categories at the pinned structured revision, records the same incoming-comma convention and the dual. Section 2 supplies the full comparison proof.
• Colimits as connected components, Sections 1–3, supplies the full small Set quotient factors, the exact element-component bijection, the constant-singleton component map and the covariant representable identity-class calculation.
• Compatible families and universal cones, Sections 2–3, supplies full cone and cocone value functors, actual universal structural maps, restriction, opposite-category reversal and size transfer. Points and representations, Section 2 supplies complete covariant Yoneda and reflection of invertible represented comparisons.
• Universes and small categories, Sections 1–4, supplies the complete Cantor, image, union and category-data bounds. Zero maps, components and subobjects, Sections 1 and 4, supplies the complete comma laws and finite-zigzag connectedness proofs.