# Kerodon

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### 7.3.6 The Universal Property of Kan Extensions

The goal of this section is to show that Kan extensions (when they exist) can be characterized by a universal mapping property.

Proposition 7.3.6.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be diagrams, and let $\beta : F_0 \rightarrow F \circ \delta$ be a natural transformation which exhibits $F$ as a left Kan extension of $F_0$ along $\delta$. Then, for every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}( F_0, G \circ \delta )$

is a homotopy equivalence of Kan complexes.

We will give the proof of Proposition 7.3.6.1 at the end of this section.

Warning 7.3.6.2. In classical category theory, some authors take the universal property of Proposition 7.3.6.1 as the definition of a Kan extension. Beware that this is a slightly different notion in general: it is possible for a natural transformation $\beta : F_0 \rightarrow F \circ \delta$ to satisfy the universal property of Proposition 7.3.6.1 without exhibiting $F$ as a left Kan extension of $F_0$ along $\delta$ (in which case $F_0$ cannot admit any other left Kan extension along $\delta$; see Corollary 7.3.6.5).

Corollary 7.3.6.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories, and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that every diagram $F_0: K \rightarrow \operatorname{\mathcal{D}}$ has a left Kan extension along $\delta$. Then the restriction functor

$\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \circ \delta } \operatorname{Fun}(K, \operatorname{\mathcal{D}})$

has a left adjoint, which carries each diagram $F_0: K \rightarrow \operatorname{\mathcal{D}}$ to a left Kan extension of $F_0$ along $\delta$.

Corollary 7.3.6.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{D}}$ admits colimits indexed by the simplicial set $K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Then the restriction functor

$\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \circ \delta } \operatorname{Fun}(K, \operatorname{\mathcal{D}})$

has a left adjoint, which carries each diagram $F_0: K \rightarrow \operatorname{\mathcal{D}}$ to a left Kan extension of $F_0$ along $\delta$.

Corollary 7.3.6.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$, and suppose that $F_0$ admits a left Kan extension along $\delta$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\beta : F_0 \rightarrow F \circ \delta$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta$.

$(2)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}( F_0, G \circ \delta )$

is a homotopy equivalence of Kan complexes.

$(3)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(K, \operatorname{\mathcal{D}})} }( F \circ \delta , G \circ \delta )) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta )$

is a bijection of sets.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.1 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3) \Rightarrow (1)$. By assumption, there exists a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta ': F_0 \rightarrow F' \circ \delta$ which exhibits $F'$ as a left Kan extension of $F$ along $\delta$. Applying Proposition 7.3.6.1, we deduce that there exists a natural transformation $\gamma : F' \rightarrow F$ for which $\beta$ is a composition of $\beta '$ with the induced transformation $\gamma |_{K}: (F' \circ \delta ) \rightarrow (F \circ \delta )$. For each object $G \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \ar [rr]^{ \circ [\gamma ] } \ar [dr] & & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F', G ) \ar [dl] \\ & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ), & }$

where the right vertical map is bijective. If condition $(3)$ is satisfied, then the left vertical map is also bijective. Allowing the functor $G$ to vary, it follows that the homotopy class $[\gamma ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }$, so that $\gamma$ is an isomorphism in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Invoking Remark 7.3.1.12, we conclude that $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta$. $\square$

Remark 7.3.6.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$. It follows from Corollary 7.3.6.5 that if $F_0$ admits a left Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ along $\delta$, then the isomorphism class of the functor $F$ is uniquely determined: it is characterized by the requirement that it corepresents the functor

$\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})} \rightarrow \operatorname{Set}\quad \quad G \mapsto \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ).$

We will deduce Proposition 7.3.6.1 from a more general statement about relative Kan extensions.

Proposition 7.3.6.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors having restrictions $F_0 = F|_{\operatorname{\mathcal{C}}^{0}}$ and $G_0 = G|_{\operatorname{\mathcal{C}}^{0}}$, so that we have a commutative diagram of Kan complexes

7.28
$$\begin{gathered}\label{equation:relative-Kan-extension-mapping-property} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ G) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( U \circ F_0, U \circ G_0). } \end{gathered}$$

If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ or $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then (7.28) is a homotopy pullback square.

Remark 7.3.6.8. In the situation of Proposition 7.3.6.7, the horizontal maps in the diagram (7.28) are Kan fibrations (Corollary 4.1.4.2 and Proposition 4.6.1.21). Consequently, the diagram (7.28) is a homotopy pullback square if and only if the induced map

$\xymatrix@C =50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [d]^{\theta } \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0) \times _{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( UF_0 , UG_0) } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( UF, UG)}$

is a homotopy equivalence (Example 3.4.1.3). Writing $\operatorname{\mathcal{M}}$ for the fiber product

$\operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$

and $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{M}}$ for the functor given by $V(H) = ( H|_{\operatorname{\mathcal{C}}^{0}}, U \circ H)$, we can identify $\theta$ with the map $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{M}}}( V(F), V(G) )$ determined by $V$. We can therefore restate Proposition 7.3.6.7 as follows:

• If the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-initial when viewed as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

• If the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-final when viewed as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Proof of Proposition 7.3.6.7. We will assume that the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (the proof in the case where $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ is similar). Using Corollary 4.5.2.23, we can factor the functor $U$ as a composition $\operatorname{\mathcal{D}}\xrightarrow {T} \operatorname{\mathcal{D}}' \xrightarrow {U'} \operatorname{\mathcal{E}}$, where $U'$ is an isofibration and $T$ is an equivalence of $\infty$-categories. Note that the functor $T \circ F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.15), and that the natural maps

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}') }( T \circ F, T \circ G)$

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})}( F_0, G_0 ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}') }( T \circ F_0, T \circ G_0)$

are homotopy equivalences. Consequently, we can replace $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{D}}'$ and thereby reduce to proving Proposition 7.3.6.7 in the special case where the functor $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is an isofibration of $\infty$-categories.

Let $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be as in Remark 7.3.6.8; we wish to show that $F$ is a $V$-initial object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Note that $V$ is also an isofibration (Proposition 4.4.5.1). By virtue of Corollary 7.1.4.17, it will suffice to show that every lifting problem

7.29
$$\begin{gathered}\label{equation:relative-Kan-extension-mapping} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d]^{V} \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) } \end{gathered}$$

has a solution, provided that $n \geq 0$ and $\sigma _0(0) = F$. Unwinding the definitions, we can rewrite (7.29) as a lifting problem

7.30
$$\begin{gathered}\label{equation:relative-Kan-extension-mapping2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^0 \ar [r]^-{ G_0 } \ar [d] & \operatorname{Fun}(\Delta ^ n, \operatorname{\mathcal{D}}) \ar [d]^{V'} \\ \operatorname{\mathcal{C}}\ar@ {-->}[ur]^{G} \ar [r] & \operatorname{Fun}(\operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\partial \Delta }^ n, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{E}}) } \end{gathered}$$

Note that $V'$ is also an isofibration of $\infty$-categories (Proposition 4.4.5.1).

We will complete the proof by showing that the lifting problem (7.30) admits a solution $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\Delta ^ n, \operatorname{\mathcal{D}})$ which is $V'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. By virtue of Proposition 7.3.5.5, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the induced lifting problem

7.31
$$\begin{gathered}\label{equation:relative-Kan-extension-mapping3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^0_{/C} \ar [r] \ar [d] & \operatorname{Fun}(\Delta ^ n, \operatorname{\mathcal{D}}) \ar [d]^{V'} \\ ( \operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \ar@ {-->}[ur]^{Q} \ar [r] & \operatorname{Fun}(\operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\partial \Delta }^ n, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{E}}) } \end{gathered}$$

admits a solution $Q: (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$ which is a $V'$-colimit diagram. Our assumption that $\sigma _0(0) = F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ guarantees that the composite map

$( \operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) = \operatorname{\mathcal{D}}$

is a $U$-colimit diagram. Applying Corollary 7.1.6.6, we conclude that the lifting problem (7.31) admits a solution $Q$, and Proposition 7.1.6.9 guarantees that $Q$ is automatically a $V'$-colimit diagram. $\square$

Corollary 7.3.6.9. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or $G$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then the restriction map

$\theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } )$

is a trivial Kan fibration.

Proof. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Assume either that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or that $G$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$. Applying Proposition 7.3.6.7 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, we deduce that the restriction map

$\theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } )$

is a homotopy equivalence of Kan complexes. Since the restriction map $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$ is an inner fibration of $\infty$-categories (Corollary 4.1.4.2), the map $\theta$ is also a Kan fibration (Proposition 4.6.1.21), and therefore a trivial Kan fibration (Proposition 3.3.7.6). $\square$

Corollary 7.3.6.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing an initial object $C$, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration of $\infty$-categories, and let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a pair of functors satisfying $U \circ F = U \circ G$. Suppose that $F$ carries each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$. Then evaluation at $C$ induces a homotopy equivalence

$\theta : \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( \{ C\} , \operatorname{\mathcal{D}}) }( F(C), G(C)).$

Proof. Let $\operatorname{\mathcal{C}}^{\mathrm{init}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its initial objects. The morphism $\theta$ then factors as a composition

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) & \xrightarrow {\theta '} & \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}^{\mathrm{init}}, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{\mathrm{init}}}, G|_{ \operatorname{\mathcal{C}}^{\mathrm{init}}} ) \\ & \xrightarrow {\theta ''} & \operatorname{Hom}_{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( \{ C\} , \operatorname{\mathcal{D}}) }( F(C), G(C)). \end{eqnarray*}

Our assumption guarantees that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{\mathrm{init}}$ (Corollary 7.3.3.13), so that the morphism $\theta '$ is a homotopy equivalence (Proposition 7.3.6.7). Since $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex (Corollary 4.6.7.14), the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}^{\mathrm{init}}$ is a categorical equivalence of simplicial sets, which implies that $\theta ''$ is also a homotopy equivalence. $\square$

Corollary 7.3.6.11. Let $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration. Suppose that $\operatorname{\mathcal{C}}$ contains an initial object $C$ having image $E = \overline{F}(C)$ and that the $\infty$-category $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ has an initial object. Then the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is initial if and only if it satisfies the following conditions:

$(1)$

The image $F(C)$ is an initial object of the $\infty$-category $\operatorname{\mathcal{D}}_{E}$.

$(2)$

The functor $F$ carries each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

Proof. It follows from Corollary 7.3.6.10 that any object of $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ which satisfies conditions $(1)$ and $(2)$ is initial. It will therefore suffice to show that there exists an object $F \in \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $(1)$ and $(2)$ (any other initial object of $\operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ will be isomorphic to $F$, and will therefore also satisfy $(1)$ and $(2)$).

Let $\operatorname{\mathcal{C}}^{\mathrm{init}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its initial objects, and let $D$ be an initial object of the $\infty$-category $\operatorname{\mathcal{D}}_{E}$. Since $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex, we can lift $\overline{F}|_{ \operatorname{\mathcal{C}}^{\mathrm{init}}}$ to a functor $F_0: \operatorname{\mathcal{C}}^{\mathrm{init}} \rightarrow \operatorname{\mathcal{D}}$ satisfying $F_0(C) = D$. Corollary 7.3.5.11 then guarantees that $F_0$ admits a $U$-left Kan extension $F \in \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, which satisfies condition $(2)$ by virtue of Corollary 7.3.3.13. $\square$

Note that relative Kan extensions are characterized by the mapping property described in Proposition 7.3.6.7:

Corollary 7.3.6.12. Suppose we are given a commutative diagram of $\infty$-categories

7.32
$$\begin{gathered}\label{equation:characterize-relative-Kan} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{F} } \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered}$$

where $\operatorname{\mathcal{C}}^{0}$ is a full subcategory of $\operatorname{\mathcal{C}}$. Assume that the lifting problem (7.32) admits a solution given by a functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an arbitrary solution to the lifting problem (7.32). Then the following conditions are equivalent:

$(1)$

The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} }) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ G) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( U \circ F|_{\operatorname{\mathcal{C}}^{0} }, U \circ G|_{ \operatorname{\mathcal{C}}^{0} }). }$

is a homotopy pullback square.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.7. To prove the converse, let $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a solution to the lifting problem (7.32) which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be as in Remark 7.3.6.8. If condition $(2)$ is satisfied, then $F$ and $F'$ are both $V$-initial objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $V(F) = V(F')$. Applying Corollary 7.1.4.12, we see that $F$ and $F'$ are isomorphic as objects of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, so that $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.17). $\square$

Corollary 7.3.6.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $F_0 = F|_{\operatorname{\mathcal{C}}^{0}}$ be the restriction of $F$ to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Suppose that the functor $F_0$ admits a left Kan extension to $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the restriction map

$\theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})}( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } )$

is a homotopy equivalence of Kan complexes.

$(3)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the restriction map

$\theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})}( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } )$

is a trivial Kan fibration of simplicial sets.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows by applying Corollary 7.3.6.12 in the special case $\operatorname{\mathcal{E}}= \Delta ^{0}$. The equivalence $(2) \Leftrightarrow (3)$ is a special case of Proposition 3.3.7.6, since the morphism $\theta$ is automatically a Kan fibration (see Corollary 4.1.4.2 and Proposition 4.6.1.21). $\square$

Combining Proposition 7.3.6.7 with the existence criterion of Proposition 7.3.5.5, we obtain the following:

Theorem 7.3.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories. Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $\operatorname{\mathcal{B}}$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ whose objects correspond to lifting problems

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}}$

with the following property:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$, the induced lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0}_{/C} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ ( \operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright } \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}}$

admits a solution which is a $U$-colimit diagram $( \operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$.

Then the restriction map

$V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$

restricts to a trivial Kan fibration $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$.

Stated more informally, Theorem 7.3.6.14 asserts that if we are given a lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur]^{F} & \operatorname{\mathcal{E}}}$

which has a possibility to be solved by a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then the functor $F$ exists and is unique up to a contractible space of choices.

Proof of Theorem 7.3.6.14. Note that the functor $V$ is an isofibration of $\infty$-categories (Proposition 4.4.5.1). It follows from Proposition 7.3.5.5 that $\operatorname{\mathcal{B}}$ is the essential image of the functor $V|_{ \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}$, and from Proposition 7.3.6.7 (together with Remark 7.3.6.8) that every object of $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is $V$-initial when regarded as an object of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Applying Corollary 7.1.4.18, we see that the functor $V|_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }: \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is a trivial Kan fibration. $\square$

Corollary 7.3.6.15. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories and let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $\operatorname{Fun}'( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ spanned by those functors $F_0$ which satisfy the following condition:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$, the diagram

$\operatorname{\mathcal{C}}^{0}_{/C} = \operatorname{\mathcal{C}}^0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}^{0} \xrightarrow {F_0} \operatorname{\mathcal{D}}$

has a colimit in the $\infty$-category $\operatorname{\mathcal{D}}$.

Then the restriction map $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ is a trivial Kan fibration of simplicial sets.

Proof. Apply Theorem 7.3.6.14 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Proof of Proposition 7.3.6.1. Let $F, G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty$-categories. Suppose we are given a simplicial set $K$ equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$, together with a natural transformation $\beta : F_0 \rightarrow F \circ \delta$ which exhibits $F$ as a left Kan extension of $F_0$ along $\delta$. Let $\theta$ denote the composite map

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}( F_0, G \circ \delta ).$

We wish to show that $\theta$ is a homotopy equivalence.

It follows from Corollary 4.1.3.3 that there exists an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty$-category. Since $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty$-categories, we can extend $\delta$ and $F_0$ to functors $\delta ': \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ and $F'_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, respectively (Proposition 1.5.6.7). Moreover, the restriction functor $\operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{D}})$ is a trivial Kan fibration (Proposition 1.5.7.6). We can therefore extend $\beta$ to a natural transformation $\beta ': F'_0 \rightarrow F \circ \delta '$, which induces a map of Kan complexes $\theta ': \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F'_0, G \circ \delta ' )$. By construction, the map $\theta$ is obtained (up to homotopy) by composing $\theta '$ with the restriction map $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F'_0, G \circ \delta ' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta )$, which is a trivial Kan fibration. Consequently, to show that $\theta$ is a homotopy equivalence, it will suffice to show that $\theta '$ is a homotopy equivalence. We may therefore replace $K$ by $\operatorname{\mathcal{K}}$ and thereby reduce to proving Proposition 7.3.6.1 in the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty$-category.

Let $\overline{\operatorname{\mathcal{C}}}$ denote the relative join $\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. Note that the definition of $\theta$ (as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$) depends only on the homotopy class of $\beta$. We may therefore assume without loss of generality that there exists a functor $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ for which $\overline{F}|_{\operatorname{\mathcal{K}}} = F_0$, $\overline{F}|_{\operatorname{\mathcal{C}}} = F$, and the natural transformation $\beta$ is given by the composition

$\Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {\overline{F}} \operatorname{\mathcal{D}}.$

Let $\overline{G}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ denote the functor given by the composition

$\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\simeq \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{D}}.$

Our assumption on $\beta$ guarantees that $\overline{F}$ is left Kan extended from the full subcategory $\operatorname{\mathcal{K}}\subseteq \overline{\operatorname{\mathcal{C}}}$ (Proposition 7.3.2.11). Applying Corollary 7.3.6.9, we deduce that precomposition with the inclusion $\operatorname{\mathcal{K}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$ determines a trivial Kan fibration

$\varphi _{-}: \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta ).$

We claim that $\overline{G}$ is right Kan extended from the full subcategory $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$. To prove this, it will suffice to show that for every object $X \in \operatorname{\mathcal{K}}$, the functor $\overline{G}$ is right Kan extended from $\operatorname{\mathcal{C}}$ at $X$ (see Proposition 7.3.3.7). Let $e_{X}: X \rightarrow \delta (X)$ denote the morphism in $\overline{\operatorname{\mathcal{C}}}$ given by the edge

$\Delta ^1 \simeq \{ X\} \star _{ \{ \delta (X) \} } \{ \delta (X) \} \hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}= \overline{\operatorname{\mathcal{C}}}.$

Note that $e_{X}$ is cocartesian with respect to the projection map $\overline{\operatorname{\mathcal{C}}} \rightarrow \Delta ^1$ (Proposition 5.2.3.15), and therefore exhibits $\delta (X)$ as a $\operatorname{\mathcal{C}}$-reflection of $X$ in the $\infty$-category $\overline{\operatorname{\mathcal{C}}}$ (Lemma 6.2.3.1). It will therefore suffice to show that $\overline{G}$ carries $e_{X}$ to an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$, which is clear (by construction, $\overline{G}(e_ X)$ is the identity morphism $\operatorname{id}_{D}$ for $D = G( \delta (X) )$). Applying Corollary 7.3.6.9 again, we deduce that precomposition with the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$ determines a trivial Kan fibration

$\varphi _{+}: \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G).$

Let $\varphi _{\pm }: \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}})}( F \circ \delta , G \circ \delta )$ be given by precomposition with the functor $\operatorname{\mathcal{K}}\xrightarrow {\delta } \operatorname{\mathcal{C}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$. Consider the diagram of Kan complexes

7.33
$$\begin{gathered}\label{equation:mapping-property-of-Kan} \xymatrix@R =50pt@C=45pt{ & \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \ar [dl]_{\varphi _{+}} \ar [d]^{\varphi _{\pm }} \ar [dr]^{\varphi _{-}} & \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})}( F \circ \delta , G \circ \delta ) \ar [r]^-{ \circ [\beta ] } & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta ). } \end{gathered}$$

Note that the diagonal maps are homotopy equivalences, and the triangle on the left is commutative. Consequently, to show that $\theta$ is a homotopy equivalence, it will suffice to show that the triangle on the right commutes up to homotopy.

Let $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta )$ be the Kan complex introduced in Notation 4.6.9.1. To verify the homotopy commutativity of the right triangle in the diagram (7.33), it will suffice to show that there is exists map of Kan complexes $\rho : \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta )$ satisfying the following conditions:

• The composition

$\operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta )$

is the constant map taking the value $\beta$.

• The composition

$\operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta )$

is equal to $\varphi _{-}$.

• The composition

$\operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta )$

is equal to $\varphi _{\pm }$.

Let $\sigma$ denote the $2$-simplex of $\Delta ^1 \times \Delta ^1$ given on vertices by the formulae

$\sigma (0) = (0,0) \quad \quad \sigma (1) = (0,1) \quad \quad \sigma (2) = (1,1),$

and let $T: \Delta ^2 \times \operatorname{\mathcal{K}}\rightarrow \Delta ^1 \times \overline{\operatorname{\mathcal{C}}}$ be the functor given by the composition

\begin{eqnarray*} \Delta ^2 \times \operatorname{\mathcal{K}}& \xrightarrow {\sigma \times \operatorname{id}_{\operatorname{\mathcal{K}}} } & \Delta ^1 \times \Delta ^1 \times \operatorname{\mathcal{K}}\\ & \simeq & \Delta ^1 \times ( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}) \\ & \rightarrow & \Delta ^1 \times ( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \\ & = & \Delta ^1 \times \overline{\operatorname{\mathcal{C}}}. \end{eqnarray*}

More concretely, the functor $T$ is given on objects by the formulae

$T(0,X) = (0,X) \quad \quad T( 1,X ) = (0, \delta (X) ) \quad \quad T(2,X) = (1, \delta (X) ).$

We conclude by observing that precomposition with $T$ induces a map of Kan complexes

$\rho : \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta )$

having the desired properties. $\square$