Extending diagrams by universal maps

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

Restricting a diagram forgets values outside the selected indices. Extending it requires a rule for those missing values and the arrows that connect them. There are two universal rules. One assembles all maps arriving at an index by a colimit; the other matches all maps leaving it by a limit. The comparison with restriction specifies what “universal” means and makes the extension coherent on every arrow.

Compose from right to left. All categories have sets of objects and arrows in ambient set theory. Choose sufficiently large ambient universes and assume ambient choice, as in Points, fibres and universal representations, Section 1. A working universe \(\mathcal U\) fixes the meaning of “small.” Functor and transformation collections may require a larger universe \(\mathcal V\); this does not turn a hypothesis about \(\mathcal U\)-small colimits into one about all \(\mathcal V\)-small colimits.

Keep the complete precomposition and component-inverse proofs in Natural transformations, Sections 1–2, the complete adjunction proofs in Passing maps across an adjunction, Sections 1–3, and the full cone/cocone and opposite-category proofs in Compatible families, Sections 2–4. The constructions below use actual universal families, including empty ones.

1. Restriction and the two universal extensions

Fix \(\phi:J\to I\). Write \(\mathcal A=\operatorname{Fun}(I,C)\), \(\mathcal B=\operatorname{Fun}(J,C)\), and \(\operatorname{Nat}_I,\operatorname{Nat}_J\) for their Hom sets. The restriction functor is

The full retained precomposition proof verifies naturality of these components and preservation of identities and vertical composition. Thus the operation restricts both diagrams and maps between them.

A left Kan extension functor \(L=\operatorname{Lan}_\phi\) is a left adjoint of \(r\). A right Kan extension functor \(R=\operatorname{Ran}_\phi\) is a right adjoint of \(r\). They have bijections

natural in both \(\beta\in\mathcal B\) and \(D\in\mathcal A\). An arbitrary set of objectwise bijections does not define an adjunction without this naturality.

Let \(\eta:1_{\mathcal B}\to rL\) and \(\zeta:Lr\to1_{\mathcal A}\) be the left unit and counit; let \(\nu:1_{\mathcal A}\to Rr\) and \(\varepsilon:rR\to1_{\mathcal B}\) be the right unit and counit. The two transpositions in (1.2), in the displayed left-to-right direction, are

Their inverses send \(u:\beta\to rD\) to \(\zeta_D L(u)\), and \(w:rD\to\beta\) to \(R(w)\nu_D\). These are the exact general adjunction formulas of the retained proof; they include naturality of all four transformations and the triangle equations

In Sections 3–4 we construct the first and last of these four universal transformations directly. For those pointwise constructions, \(\zeta_D\) is the unique map induced at \(i\) by the family \(D(t):D(\phi j)\to D(i)\), where \(t:\phi j\to i\). The component \(\nu_{D,i}\) has projection \(D(t):D(i)\to D(\phi j)\), where \(t:i\to\phi j\). Thus both other transformations have specified maps as well.

2. One diagram or an entire adjoint

An individual diagram \(\beta:J\to C\) can have an extension even when \(r\) has no adjoint on every diagram. A left extension is an object \(L\beta\in\mathcal A\) representing the covariant functor \(D\mapsto\operatorname{Nat}_J(\beta,rD)\):

The specified correspondence is natural in \(D\). Its universal element is \(\eta_\beta:\beta\to rL\beta\), and the forward map is \(h\mapsto(rh)\eta_\beta\). In this representation, the arrows go out of the representing object. A right extension represents the contravariant functor \(D\mapsto\operatorname{Nat}_J(rD,\beta)\):

Its universal element is \(\varepsilon_\beta:rR\beta\to\beta\), and the forward map is \(v\mapsto\varepsilon_\beta(rv)\). Here the arrows go into the representative.

These characterizations use the complete covariant and contravariant universal-element proofs in Solution sets and universal representations, Section 1. They also give unique isomorphisms between two extensions commuting with their specified universal transformations. An abstract isomorphism of their underlying diagrams does not specify that comparison.

If (2.2) holds for every \(\beta\), retain the full represented-Hom-family construction in Points and representations, Section 4, applied to \(r:\mathcal A\to\mathcal B\). It constructs \(R\) on arrows from the chosen representations, proves both functor laws and both-variable naturality, and proves compatible uniqueness. Apply that complete result to \(r^{\mathrm{op}}\) for (2.1): it constructs \(L^{\mathrm{op}}\), hence \(L\). This includes empty functor categories and uses choices indexed by ambient sets. Hom encodings and representation comparisons after universe enlargement are exactly those in the retained Sections 1 and 7 of that lesson. The object category \(C\) remains fixed.

A Kan extension defined by (2.1) or (2.2) is a universal object in a functor category. The next formulas are sufficient constructions using specified pointwise (co)limits; their existence is a hypothesis, rather than an assertion that every universal extension must have these pointwise descriptions without further conditions.

3. Build the left extension

For \(i\in I\), let \(K_i=(\phi\downarrow i)\). An object is \((j,t:\phi(j)\to i)\); an arrow to \((j',t')\) is \(s:j\to j'\) with \(t'\phi(s)=t\). The complete comma-category proof in Zero maps, Section 1 gives its category laws and the forgetful functor \(K_i\to J\). Compose it with \(\beta\) to get the diagram with value \(\beta(j)\) and arrow \(\beta(s)\).

Theorem 3.1. If this diagram has a colimit in \(C\) for every \(i\), then \(\beta\) has a left Kan extension. No smallness of \(I\) or \(J\) is needed for this conditional statement.

Choose its universal cocones and write

The cocone equation is \(\lambda^i_{j',t'}\beta(s)=\lambda^i_{j,t}\) whenever \(t'\phi(s)=t\).

For \(a:i\to i'\), postcomposition sends \(K_i\) to \(K_{i'}\) by \((j,t)\mapsto(j,at)\), and keeps its arrows \(s\). It preserves the triangle equation, identities and compositions. The maps \(\lambda^{i'}_{j,at}\) form a cocone on \(K_i\), so they induce the unique arrow

For \(a=1_i\), the identity has these composites. For \(a':i'\to i''\), the composite \((L\beta)(a')(L\beta)(a)\) has composite \(\lambda^{i''}_{j,a'at}\) with each \(\lambda^i_{j,t}\), so it is \((L\beta)(a'a)\). Colimit uniqueness proves both laws and makes \(L\beta\) a functor.

Define \(\eta_{\beta,j}=\lambda^{\phi j}_{j,1_{\phi j}}\). For \(s:j\to j'\), there is an arrow \(s:(j,\phi s)\to(j',1_{\phi j'})\) in \(K_{\phi j'}\). Its cocone equation and (3.2) give

Hence \(\eta_\beta:\beta\to rL\beta\) is natural.

For a transformation \(z:\beta\to\beta'\) between diagrams with these pointwise colimits, the family \(\lambda'{}^i_{j,t}z_j\) is a cocone: naturality of \(z\) and the primed cocone equation verify every comma arrow. It gives the unique \((Lz)_i\) with \((Lz)_i\lambda^i_{j,t}=\lambda'{}^i_{j,t}z_j\). Testing these composites proves naturality in \(i\) by (3.2). The same test proves \(L(1_\beta)=1_{L\beta}\) and \(L(z'z)=L(z')L(z)\). At \((j,1)\) it gives \((rLz)_j\eta_{\beta,j}=\eta_{\beta',j}z_j\), which is naturality of the unit in \(\beta\).

To prove universality, take \(u:\beta\to rD\). For fixed \(i\), the maps \(D(t)u_j:\beta(j)\to D(i)\) form a cocone. Indeed, for \(t'\phi(s)=t\),

Let \(h_i:(L\beta)(i)\to D(i)\) be its unique factor. For \(a:i\to i'\), both \(D(a)h_i\) and \(h_{i'}(L\beta)(a)\) compose with \(\lambda^i_{j,t}\) to \(D(at)u_j\). Uniqueness makes \(h:L\beta\to D\) natural.

Conversely, a natural \(h:L\beta\to D\) gives \(u_j=h_{\phi j}\eta_{\beta,j}\). Naturality of \(h\) at \(\phi(s)\) and (3.3) give \(D(\phi s)u_j=u_{j'}\beta(s)\), so \(u\) is natural. Starting from \(u\), the value of its factor at \((j,1)\) is \(h_{\phi j}\eta_{\beta,j}=D(1)u_j=u_j\), giving one inverse equation. Starting from \(h\), its reconstructed factor has composite

at every coprojection. Colimit uniqueness gives the original \(h_i\). This proves the other inverse equation, including empty \(K_i\).

The correspondence is natural in both variables. For \(z:\beta'\to\beta\), \(v:D\to D'\), its forward map \(\Psi(h)=(rh)\eta_\beta\) satisfies

This follows from naturality of \(\eta\) in \(\beta\) just proved and the restriction functor laws. A family of bijections whose forward maps obey these squares has inverse maps obeying them too. Thus the complete factors prove (2.1), and when the hypotheses hold for every \(\beta\), they give \(L\dashv r\). \(\square\)

4. Build the right extension

Let \(H_i=(i\downarrow\phi)\). Its objects are \((j,t:i\to\phi(j))\); an arrow to \((j',t')\) is \(s:j\to j'\) with \(\phi(s)t=t'\). Retain the complete dual comma-category construction and forgetful functor from Zero maps, Section 1.

Theorem 4.1. If \(\beta:H_i\to C\), via this forgetful functor, has a limit for every \(i\), then \(\beta\) has a right Kan extension. This conditional statement again allows arbitrary ambient \(I,J\).

Choose the limits and write

The cone equations are \(\beta(s)p^i_{j,t}=p^i_{j',t'}\) whenever \(\phi(s)t=t'\).

An arrow \(a:i\to i'\) sends \(H_{i'}\) to \(H_i\) by \((j,t')\mapsto(j,t'a)\), keeping \(s\). The triangle equations and category laws are preserved. The maps \(p^i_{j,t'a}\), indexed by \(H_{i'}\), form a cone and give the unique arrow characterized by

For an identity the identity satisfies this equation. For successive \(a,a'\), testing at each projection in \(H_{i''}\) shows that \((R\beta)(a')(R\beta)(a)\) and \((R\beta)(a'a)\) have the same projections \(p^i_{j,t''a'a}\). Uniqueness proves composition, so \(R\beta\) is a functor.

A transformation \(z:\beta\to\beta'\) gives a cone \(z_jp^i_{j,t}\), because the naturality of \(z\) intertwines the two cone equations. Its unique factor \((Rz)_i\) satisfies \(p'{}^i_{j,t}(Rz)_i=z_jp^i_{j,t}\). Testing at the target projections proves naturality in \(i\) using (4.2), and proves preservation of identities and compositions of \(z\).

Define \(\varepsilon_{\beta,j}=p^{\phi j}_{j,1_{\phi j}}\). For \(s:j\to j'\), the arrow \(s:(j,1)\to(j',\phi s)\) in \(H_{\phi j}\), together with (4.2), gives

This proves naturality in \(j\); the transformation-factor equation at \((j,1)\) proves naturality of \(\varepsilon\) in \(\beta\).

For \(w:rD\to\beta\), the maps \(w_jD(t):D(i)\to\beta(j)\) give a cone on \(H_i\). For its arrow \(s\), naturality of \(w\) gives \(\beta(s)w_jD(t)=w_{j'}D(\phi s)D(t)=w_{j'}D(t')\). Its unique factor \(v_i:D(i)\to(R\beta)(i)\) is natural in \(i\): for \(a:i\to i'\), the two arrows \((R\beta)(a)v_i\), \(v_{i'}D(a)\) have the same projection \(w_jD(t'a)\) at every \((j,t')\in H_{i'}\).

Conversely, a natural \(v:D\to R\beta\) gives \(w_j=\varepsilon_{\beta,j}v_{\phi j}\). Naturality of \(v\) at \(\phi(s)\), followed by (4.3), proves that \(w\) is natural. Starting from \(w\), the factor at \((j,1)\) gives \(\varepsilon_{\beta,j}v_{\phi j}=w_jD(1)=w_j\). Starting from \(v\), the reconstructed factor has projection

Limit uniqueness therefore gives the original \(v_i\). Both inverse equations hold even when \(H_i\) is empty.

For \(z:\beta\to\beta'\), \(b:D'\to D\), the forward map \(\Xi(v)=\varepsilon_\beta(rv)\) obeys

Use naturality of \(\varepsilon\) in \(\beta\) and the restriction functor laws. This proves both-variable naturality; the inverse maps commute as well because these are bijections. We have proved (2.2) with every factor specified. When the hypotheses hold for all \(\beta\), this gives \(r\dashv R\). \(\square\)

5. Recover the values on a full embedding

For the usual small-diagram consequence, suppose \(J\) is \(\mathcal U\)-small and \(I\) is locally \(\mathcal U\)-small. This is the working-universe convention for the target index category; its object set may be larger. The objects of \(K_i\) form a tagged union of the small sets \(I(\phi j,i)\) over the small object set of \(J\). By the complete transported-union proof in Universes and small categories, Section 3, that union is small. For each pair of these objects, its Hom set is a subset of a Hom set of \(J\); the union of those subsets over the small set of pairs is small too. Thus \(K_i\) is a small category. The proof for \(H_i\) uses \(I(i,\phi j)\) and the same closure arguments.

Consequently, if \(C\) has all \(\mathcal U\)-small colimits, Theorem 3.1 applies to every \(\beta\); if it has all \(\mathcal U\)-small limits, Theorem 4.1 applies to every \(\beta\). Choices are indexed by ambient objects, and the already proved arrow rules or the full representative construction of Section 2 make the adjoints coherent. The size proof did not require \(I\) to be small. If a conditional comma diagram is larger than \(\mathcal U\), Sections 3–4 still apply when its needed colimit or limit exists; the all-small consequence alone does not supply that larger object.

Now suppose \(\phi\) is fully faithful. In \(K_{\phi j}\), the object \((j,1_{\phi j})\) is terminal. From \((j',t)\) to it, the comma arrow is the unique \(s:j'\to j\) with \(\phi(s)=t\), supplied by full faithfulness. The complete terminal-index colimit proof in Universal forks, Section 1 identifies the colimit with \(\beta(j)\). The specified comparison is precisely \(\eta_{\beta,j}\), so it is invertible.

For completeness, its inverse \(c:(L\beta)(\phi j)\to\beta(j)\) has \(c\lambda_{j',t}=\beta(s)\), where \(\phi(s)=t\). These form a cocone by the faithful functor law. At \((j,1)\), \(c\eta_{\beta,j}=1\). At any \((j',t)\), the cocone equation gives \(\eta_{\beta,j}\beta(s)=\lambda_{j',t}\); hence \(\eta_{\beta,j}c\) has the same composites as the identity with every coprojection. Uniqueness gives the other inverse equation.

In \(H_{\phi j}\), \((j,1_{\phi j})\) is initial: the unique arrow to \((j',t)\) is the unique \(s:j\to j'\) with \(\phi(s)=t\). A cone from \(\beta(j)\) has components \(\beta(s)\). Its unique factor \(d:\beta(j)\to(R\beta)(\phi j)\) satisfies \(\varepsilon_{\beta,j}d=1\). The cone equation gives \(\beta(s)\varepsilon_{\beta,j}=p_{j',t}\), so \(d\varepsilon_{\beta,j}\) has each identity projection and is the identity. Thus the specified counit component is invertible.

Whenever the corresponding global adjoint exists, the component-inverse theorem makes \(\eta\), or \(\varepsilon\), a natural isomorphism. The complete unit/counit full-faithfulness criterion in Passing maps across an adjunction, Section 3, applied to \(L\dashv r\) or \(r\dashv R\), proves

respectively. Even for an individual \(\beta\) with only its conditional pointwise construction available, the particular \(\eta_\beta\) or \(\varepsilon_\beta\) above is invertible. Full faithfulness of a global functor is a claim made when that functor exists.

If \(K_i\) is empty, its colimit value is initial; if \(H_i\) is empty, its limit value is terminal, by the complete empty-diagram proof. For empty \(J\), all these comma categories are empty. The left extension is the constant initial diagram and the right extension the constant terminal diagram whenever those objects exist. Every arrow of a constant such diagram is its unique endomorphism, hence its identity, verifying the functor laws at this boundary. For empty \(I\), a functor \(J\to I\) forces \(J\) to be empty; both functor categories then have exactly one object and one arrow, and both adjunctions and comparisons are the unique ones.

6. Reverse both directions

There is an isomorphism

It sends \(D\) to \(D^{\mathrm{op}}\). An arrow \(D\to D'\) in the left category is a transformation \(D'\to D\) in the original functor category; take the opposite of each component to get \(D^{\mathrm{op}}\to D'{}^{\mathrm{op}}\). Its naturality equation is the reversed original equation. Identity and composition laws are reversed componentwise; repeating the operation is its inverse. This is the entire opposite-functor-category construction retained from Compatible families, Section 4.

On both objects and components, \(E_J r_\phi^{\mathrm{op}}=r_{\phi^{\mathrm{op}}}^{C^{\mathrm{op}}}E_I\). An adjunction \(L\dashv r\) reverses to \(r^{\mathrm{op}}\dashv L^{\mathrm{op}}\), because reversing a Hom set in the defining natural correspondence interchanges its two arguments. Transporting through \(E_I,E_J\) therefore gives a right Kan extension along \(\phi^{\mathrm{op}}\) in \(C^{\mathrm{op}}\). Compatible uniqueness of adjoints identifies

These are the comparisons respecting the specified universal maps. The left unit becomes the right counit under reversal; the left counit becomes the right unit. The equations of Section 1 reverse into the corresponding equations. The same argument applies to individual representations in Section 2.

Concretely, \((\phi\downarrow i)^{\mathrm{op}}\) is the category \((i\downarrow\phi^{\mathrm{op}})\) in the reversed indices: the arrow \(\phi j\to i\) becomes \(i\to\phi j\), and every triangle reverses. Reversing its colimit cocone gives precisely the limit cone of the right formula. This verifies the pointwise formulas and their arrow maps under the comparisons, including empty categories.

7. Total assembly and total compatibility

Suppose \(I,J\) are small and \(C\) has all small colimits, using a working universe in which the index categories and their relevant Hom sets are small. For \(\beta:J\to C\), put \(B=\operatorname{colim}_J\beta\) with coprojections \(b_j\), and \(A=\operatorname{colim}_I L\beta\) with coprojections \(c_i\). The unit gives the canonical comparison \(q:B\to A\) characterized by

It is the composite of the colimit map induced by \(\eta_\beta\) and the canonical restriction-of-index comparison for \(L\beta\). Both canonical maps are supplied by the complete structural-family proof in Compatible families, Section 2.

Proposition 7.1. This \(q\) is invertible, for every \(\phi\).

Proof. A map \(A\to T\) is exactly a cocone from \(L\beta\) to \(T\), or a transformation \(L\beta\to\Delta_I T\), by the complete cone/cocone correspondence. The constant functor sends every arrow to \(1_T\), so \(r\Delta_I T=\Delta_J T\) on all objects and arrows. The left adjunction consequently gives a natural bijection

Its action on \(h:A\to T\), using the specified unit transposition, has \(j\)-component \(h c_{\phi j}\eta_{\beta,j}=h q b_j\). Colimit uniqueness identifies this map with \(h\mapsto hq\). To see invertibility directly, take \(T=B\): the bijection gives \(a:A\to B\) with \(aq=1_B\). For \(T=A\), the maps \(qa\) and \(1_A\) have the same composite with \(q\), so injectivity gives \(qa=1_A\). Thus this exact canonical comparison is invertible. \(\square\)

The construction also proves naturality in \(\beta\): for \(z:\beta\to\beta'\), both routes from \(B\) to \(A'\), tested at \(b_j\), give \(c'_{\phi j}\eta_{\beta',j}z_j\), by unit naturality and the colimit transformation law. Universal uniqueness makes the routes equal.

There is an ordinary covariant limit form. Suppose \(C\) has all small limits. For \(\gamma:J\to C\), let \(A=\lim_I R\gamma\), \(B=\lim_J\gamma\), with projections \(p_i,b_j\). The counit gives \(q:A\to B\) with

It is the canonical index-restriction map followed by the limit map induced by the counit. The full cone property and right adjunction give, naturally in \(T\),

The action on \(h:T\to A\) has projections \(\varepsilon_{\gamma,j}p_{\phi j}h=b_j qh\), so it is postcomposition by this \(q\). At \(T=B\), surjectivity gives \(a:B\to A\) with \(qa=1_B\); at \(T=A\), injectivity gives \(aq=1_A\). Thus \(q\) is invertible. Naturality in \(\gamma\) follows by testing at every \(b_j\), using counit naturality and the retained limit transformation law.

For the projective source convention, take \(\beta:J^{\mathrm{op}}\to C\) and \(\phi^{\mathrm{op}}:J^{\mathrm{op}}\to I^{\mathrm{op}}\). The preceding limit result says

At \(i\), its inner right-Kan limit is indexed by arrows \(i\to\phi j\) in \(I^{\mathrm{op}}\), equivalently arrows \(\phi j\to i\) in \(I\). Its diagram values are \(\beta(j)\), with its reversed-\(J\) arrow action. This explains the direction in the iterated projective formula. The comparison uses the counit for \(\phi^{\mathrm{op}}\); using the counit for \(\phi\) instead would have the wrong domain.

No fully faithful hypothesis on \(\phi\) was used in either total comparison. All the universal properties and constant-diagram equalities include empty indices. When \(J\) is empty but \(I\) is not, (7.2) still identifies the total colimit of the initial-valued extension with the initial object. The dual identifies the total limit of the terminal-valued extension with the terminal object. When \(I\) is empty, \(J\) is empty too, and the same initial or terminal factor argument applies.

8. Four graded exercises

Exercise 1 — introductory: extend from either endpoint

Let \(I\) be the walking arrow \(0\to1\), \(J\) the one-object identity category, \(C=\mathsf{Set}\), and \(\beta(*)=S\). For each inclusion \(\phi_0(*)=0\), \(\phi_1(*)=1\), calculate the left and right extensions, their structural arrow, and their unit or counit. Give the universal correspondences for an arbitrary diagram \(D=(A\xrightarrow{d}B)\).

Solution. For \(\phi_0\), each left comma category has one object and its identity, using \(1_0\) at 0 and the arrow \(0\to1\) at 1. Thus \(L_0S=(S\xrightarrow{1}S)\); its unit is \(1_S\) at 0. A transformation \(L_0S\to D\) is a pair \(h_0:S\to A,h_1:S\to B\) with \(h_1=dh_0\). Choosing \(h_0\) gives exactly a map \(S\to(r_0D)=A\), with unique inverse choice \(h_1=dh_0\).

The right comma at 0 has its single identity object; the one at 1 is empty. Hence \(R_0S=(S\to1)\), where \(1\) is a singleton and the arrow is unique. The counit at 0 is \(1_S\). A transformation \(D\to R_0S\) is any \(v_0:A\to S\) together with the unique \(v_1:B\to1\). The arrow square commutes because both routes to \(1\) are unique. It is thus exactly a map \(r_0D=A\to S\), including when \(A\), \(B\) or \(S\) is empty.

For \(\phi_1\), the left comma at 0 is empty and the one at 1 has its single identity object. Thus \(L_1S=(\varnothing\to S)\), with unit \(1_S\) at 1. A transformation to \(D\) consists of the unique empty map to \(A\) and any \(h_1:S\to B\); the arrow square has empty domain and is automatic. This is exactly a map \(S\to r_1D=B\).

Both right comma categories for \(\phi_1\) have one object, so \(R_1S=(S\xrightarrow{1}S)\), with counit \(1_S\) at 1. A transformation \(D\to R_1S\) is determined by \(v_1:B\to S\), because \(v_0=v_1d\). These rules prove both right and left correspondences, all inverse equations, and the specified structural arrows. Pre- or postcomposing the free component maps commutes with the formulas, proving the required naturality.

Exercise 2 — intermediate: every arrow label matters

Let \(I\) have objects \(0,1\), their identities and two distinct arrows \(a,b:0\to1\), with no other nonidentity arrows. Include the point \(J=\{*\}\) at 0 and take \(\beta(*)=S\). Find both extensions. Describe a left-extension transformation into \(D=(A\mathrel{\substack{\xrightarrow{d_a}\\[-2pt]\xrightarrow[d_b]{}}}B)\). Does the unit recover \(S\)?

Solution. The left comma at 0 has the identity object. At 1 it has the two objects \((*,a)\), \((*,b)\), with no arrow between them: the only underlying \(J\)-arrow is the identity, which forces its two labels to agree. Hence \(LS(0)=S\), \(LS(1)=S\amalg S\), and \(LS(a)\), \(LS(b)\) are respectively its first and second inclusions. The unit at 0 is \(1_S\).

A transformation has \(h_0:S\to A\) and \(h_1:S\amalg S\to B\). Its two arrow equations force \(h_1\iota_a=d_a h_0\), \(h_1\iota_b=d_b h_0\). The coproduct property gives precisely one such \(h_1\) for every \(h_0\), with values \(d_a h_0\) and \(d_b h_0\) on the two tagged summands. Conversely evaluation at 0 recovers \(h_0\), so these are inverse correspondences. This verifies the universal extension and its full arrow data.

The right comma at 0 has the single identity object, while at 1 it is empty. Thus \(RS(0)=S\), \(RS(1)=1\), with both arrow maps the unique map to the singleton, and counit \(1_S\) at 0. A transformation \(D\to RS\) is exactly any function \(A\to S\) and the unique map \(B\to1\); both arrow squares commute by uniqueness.

The inclusion of \(J\) is fully faithful: the only Hom set it tests is \(I(0,0)=\{1_0\}\). Both unit and counit recover \(S\) at that included object, as the full embedding theorem predicts. Distinct arrows to the missing object create distinct comma labels and two left summands. If \(S=\{0,1\}\), the left value at 1 has four elements, whereas the right value there has one.

Exercise 3 — advanced: compare total colimits directly

Let \(I\) be the ordered category with objects \(0,1,2\), nonidentity arrows \(0\to2\), \(1\to2\), and no arrow between 0 and 1. Let \(J=\{0,1\}\) be discrete and include it in \(I\). For \(\beta(0)=A,\beta(1)=B\), find both extensions in sets. Compute their total colimits and total limits. Identify which two calculations are the canonical total comparisons of Section 7.

Solution. The left comma at 0 has only \((0,1)\), the one at 1 only \((1,1)\), and the one at 2 has the two objects labelled by the two arrows to 2, with no morphism between them. Thus \(L\beta\) is \(A\to A\amalg B\leftarrow B\), with the two inclusions. The right comma at 0 and at 1 have their respective identities, while the one at 2 is empty. Thus \(R\beta\) is \(A\to1\leftarrow B\), with the unique functions.

The object 2 is terminal in \(I\). The complete terminal-index factor proof gives \(\operatorname{colim}_I L\beta=A\amalg B\) and \(\operatorname{colim}_I R\beta=1\). On discrete \(J\), \(\operatorname{colim}_J\beta=A\amalg B\). Its canonical comparison to the former is the identity on both tagged summands: the units at 0,1 are identities and the outer coprojections are the inclusions.

A cone on \(L\beta\) consists of functions \(u:T\to A\), \(v:T\to B\) whose composites to \(A\amalg B\) agree. The two tagged summands are disjoint, so such a pair can exist only for empty \(T\), and then all maps are unique. Consequently \(\lim_I L\beta=\varnothing\), with its unique cone. This holds if either set is empty too.

A cone on \(R\beta\) is an arbitrary pair \(u:T\to A,v:T\to B\), together with the unique map to \(1\); both equations to \(1\) hold. Thus \(\lim_I R\beta=A\times B\), with its ordinary projections. On discrete \(J\), \(\lim_J\beta=A\times B\), and the counit comparison is the identity on both projections.

The canonical invariances of Section 7 are the total colimit of \(L\beta\) and total limit of \(R\beta\). The other two calculations give different objects and illustrate why the direction of the extension matters. All properties just computed are full factor properties, not just cardinality comparisons.

Exercise 4 — challenge: free and cofree group actions

Let \(G\) be a small group, regarded as a one-object category with composition multiplication. Include the identity-only point \(J\) and let \(\beta(*)=S\), a small set. Calculate its extensions in \(\mathsf{Set}_{\mathcal U}\). Prove both universal correspondences for every left \(G\)-set \(X\). Then compute the orbits of the left extension and the fixed elements of the right extension, and identify their canonical comparisons with \(S\).

Solution. A diagram on the group category is a set with functions \(g:X\to X\) satisfying \(g(hx)=(gh)x\) and \(1x=x\). The inverse group element gives the inverse function, so it is exactly a left \(G\)-action. Restriction forgets this action. Each comma category has one object for every \(g\in G\) and only identities, since the identity-only \(J\) supplies no nontrivial arrow. Thus \(LS=G\times S\) and \(RS=S^G\).

Postcomposition of left-comma labels gives \(h(g,s)=(hg,s)\). This is a left action because \(k(h(g,s))=((kh)g,s)\), and the identity acts trivially. Precomposition of right-comma labels gives \((h\cdot f)(g)=f(gh)\). Here \((k\cdot(h\cdot f))(g)=f(gkh)=((kh)\cdot f)(g)\), and the identity again acts trivially. The unit is \(s\mapsto(1,s)\); the counit is evaluation \(f\mapsto f(1)\). Smallness of \(G\times S\) and \(S^G\) is the retained product/function-set closure.

Given \(u:S\to X\), its unique equivariant left factor is \(H(g,s)=g\,u(s)\). Indeed \(H(hg,s)=hg\,u(s)=hH(g,s)\). Evaluating at \((1,s)\) returns \(u\). Conversely an equivariant \(H\) must satisfy \(H(g,s)=gH(1,s)\), so it is this factor. For the right correspondence, given \(w:X\to S\), define \(V(x)(g)=w(gx)\). Then \(V(hx)(g)=w(ghx)=(h\cdot V(x))(g)\), so \(V\) is equivariant. Evaluation at 1 gives \(w(x)\). Conversely an equivariant \(V\) satisfies \(V(x)(g)=(g\cdot V(x))(1)=V(gx)(1)\), forcing the same formula. Thus the two constructions and evaluations are inverse on every map. Composing \(u,w\) or the equivariant maps with a function of \(S\) or an equivariant map of \(X\) commutes with these explicit formulas, proving both-variable naturality.

Every orbit of \(G\times S\) consists of the entire fibre \(G\times\{s\}\): from \((g,s)\) to \((g',s)\) use \(g'g^{-1}\). No action changes \(s\). Hence the orbit set is \(S\), and the canonical map from \(S=\operatorname{colim}_J\beta\) sends \(s\) to the orbit of \((1,s)\), an inverse to recording \(s\). An invariant function \(f:G\to S\) satisfies \(f(gh)=f(g)\) for all \(g,h\); setting \(g=1\) makes it constant. Conversely constant functions are invariant. Their set is therefore \(S\), and the canonical comparison \(\lim_I RS\to S\) evaluates a constant function at 1, with inverse the constant-function assignment.

These calculations include empty \(S\): \(G\) is nonempty, so \(S^G\) and \(G\times S\) are then empty, and both total comparisons are the empty bijection. If \(G\) is nontrivial, the inclusion \(J\to I\) is not full, and for singleton \(S\) the left unit lands in only one of \(|G|\) elements. Thus recovery at the included object needs the full embedding hypothesis, while the total comparisons do not.

9. References and retained proof interfaces

Kan extensions, their pointwise formulas and the comma categories that compute them are treated in Emily Riehl, Category Theory in Context, Sections 6.1–6.3. Every universal map used in a comparison is specified here.

The linked course providers retain complete arguments for representing objects in diagrams, adjunctions and their triangles, the exact unit/counit full-faithfulness criterion, comma-category laws, cone/cocone factor properties, opposite functor categories and small indexing bounds. These include compatible uniqueness and the needed empty cases. Their exact specializations are stated where used.

The pinned Stacks category source supplies a full adjoint full-faithfulness criterion, already covered by the retained owned proof above. Its objectwise right-adjoint existence argument omits the functor and adjunction verification; that omission is not imported as a complete proof. The full represented-Hom-family provider, and the complete pointwise calculations of Sections 3–4, supply every such check here.