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7.3.7 Kan Extensions in Functor $\infty$-Categories

Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories (Definition 4.5.9.10). For every $\infty$-category $\operatorname{\mathcal{D}}$, Corollary 4.5.9.19 guarantees that the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ is an $\infty$-category (see Construction 4.5.9.1). The goal of this section is to describe (relative) Kan extensions of functors which take values in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. We can state our main result as follows:

Theorem 7.3.7.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $V': \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ denote the functor given by postcomposition with $V$. Let $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ be a functor of $\infty$-categories, corresponding to a morphism $\operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ and a functor $F: \operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Let $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$ be a full subcategory. If $F$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, then $f$ is $V'$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$.

We will give the proof of Theorem 7.3.7.1 at the end of this section.

Remark 7.3.7.2. In the situation of Theorem 7.3.7.1, suppose that the functor $F^{0} = F|_{ \operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}}$ admits a $V$-left Kan extension $F': \operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ satisfying $V \circ F' = V \circ F$. Then the converse of Theorem 7.3.7.1 is also true: if $f$ is $V'$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$, then $F$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$. To prove this, it will suffice to show that the functors $F$ and $F'$ are isomorphic (Remark 7.3.3.17). This is clear: we can identify $F'$ with a functor $f': \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ satisfying $V' \circ f' = V' \circ f$, and Theorem 7.3.7.1 guarantees that $f'$ is $V'$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$. Since the functors $f$ and $f'$ coincide on $\operatorname{\mathcal{A}}^{0}$, they are isomorphic by virtue of Theorem 7.3.6.14.

Corollary 7.3.7.3. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories, let $\operatorname{\mathcal{D}}$ be an $\infty$-category, and let $\pi : \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ be the projection map. Let $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ be a functor of $\infty$-categories and let $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$ be a full subcategory. If the induced map $\operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, then $f$ is $\pi$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$.

Proof. Apply Theorem 7.3.7.1 in the special case $\operatorname{\mathcal{E}}= \Delta ^{0}$. $\square$

Corollary 7.3.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $V': \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be the functor given by postcomposition with $V'$. Suppose we are given another functor $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ and a full subcategory $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$. If the induced map $\operatorname{\mathcal{A}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0} \times \operatorname{\mathcal{C}}$, then $f$ is $V'$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$.

Proof. Apply Theorem 7.3.7.1 in the special case $\operatorname{\mathcal{B}}= \Delta ^{0}$. $\square$

Corollary 7.3.7.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories and let $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ be a functor of $\infty$-categories, corresponding to a functor $F: \operatorname{\mathcal{A}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Let $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$ be a full subcategory. If $F$ is left Kan extended from $\operatorname{\mathcal{A}}^{0} \times \operatorname{\mathcal{C}}$, then $f$ is left Kan extended from $\operatorname{\mathcal{A}}^{0}$.

Proof. Apply Corollary 7.3.7.4 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Corollary 7.3.7.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, and let $V': \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ be the isofibration given by postcomposition with $V$ (see Proposition 4.5.9.18). Let $e$ be a morphism of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$, corresponding to a pair $( \overline{e}, f )$ where $\overline{e}$ is a morphism of $\operatorname{\mathcal{B}}$ and $f: \Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty$-categories. If $f$ is $V$-left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, then the morphism $e$ is $V'$-cocartesian.

Proof. Apply Theorem 7.3.7.1 in the special case $\operatorname{\mathcal{A}}= \Delta ^1$ and $\operatorname{\mathcal{A}}^{0} = \{ 0\}$ (see Example 7.1.5.9). $\square$

Example 7.3.7.7. In the situation of Corollary 7.3.7.6, suppose that $\operatorname{\mathcal{B}}= \Delta ^0$. Corollary 7.3.7.6 then asserts that a morphism $e: X \rightarrow Y$ in the functor $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is $V'$-cocartesian if, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $e_{C}: X(C) \rightarrow Y(C)$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{D}}$. This is a special case of Lemma 5.2.1.5.

Example 7.3.7.8. In the situation of Corollary 7.3.7.6, suppose that $U$ is a cartesian fibration. Let $U_{ \overline{e} }: \Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ denote the cartesian fibration given by projection onto the first factor. By virtue of Proposition 7.3.3.11, the functor $f$ is $V$-left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ if and only if it carries $U_{ \overline{e} }$-cartesian morphisms of $\Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{D}}$. In this case, Corollary 7.3.7.6 is a special case of (the dual of) Lemma 5.3.6.11.

Corollary 7.3.7.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories, let $\operatorname{\mathcal{D}}$ be an $\infty$-category, and let $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ be the projection map. Let $e$ be a morphism of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$, corresponding to a pair $( \overline{e}, f )$ where $\overline{e}$ is a morphism of $\operatorname{\mathcal{B}}$ and $f: \Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty$-categories. If $f$ is left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, then the morphism $e$ is $\pi$-cocartesian.

Proof. Apply Corollary 7.3.7.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

The proof of Theorem 7.3.7.1 will require some preliminaries.

Lemma 7.3.7.10. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, let $K$ be a simplicial set equipped with a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$, and let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ be a full subcategory. Let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$ denote the essential image of $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ under the restriction map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$. Suppose that every simplicial set $A$ satisfies the following condition:

$(\ast _{A})$

For every extension of $\overline{f}$ to a morphism $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$, the restriction functor

$\xymatrix { \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(K^{\triangleright }, \operatorname{\mathcal{D}}) } \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \ar [d]^-{ \theta _{A} } \\ \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(K, \operatorname{\mathcal{D}}) } \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) }$

is an equivalence of $\infty$-categories.

Then every object of $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is a $V$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$.

Proof. Without loss of generality, we may assume that the full subcategory $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is replete. For every extension of $\overline{f}$ to a morphism $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$, let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ denote the inverse images of $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K, \operatorname{\mathcal{D}})$, respectively. For every monomorphism of simplicial sets $A \hookrightarrow B$, Proposition 4.4.5.1 guarantees that the restriction map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ is an isofibration, and therefore induces an isofibration $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}})$ Combining Corollary 4.5.2.30 with assumptions $(\ast _{A})$ and $(\ast _{B})$, we conclude that the restriction map

$\theta _{A,B}: \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K^{\triangleright } \star A, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star A, \operatorname{\mathcal{D}}) } \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( K \star B, \operatorname{\mathcal{D}})$

is an equivalence of $\infty$-categories. Proposition 4.4.5.1 implies that $\theta _{A,B}$ is also an isofibration, and is therefore a trivial Kan fibration (Proposition 4.5.5.20). In particular, $\theta _{A,B}$ is surjective on vertices. Unwinding the definitions, we conclude that every lifting problem

$\xymatrix@R =50pt@C=50pt{ (K^{\triangleright } \star A) {\coprod }_{ (K \star A) } (K \star B) \ar [r]^-{g} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{V} \\ K^{\triangleright } \star B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}}$

admits a solution, provided that the restriction $g|_{ K^{\triangleright } }$ belongs to $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$. The desired result now follows by invoking the criterion of Remark 7.1.5.11. $\square$

Remark 7.3.7.12. In the situation of Lemma 7.3.7.10, suppose we are given an inner anodyne morphism of simplicial sets $A \hookrightarrow B$. Then every diagram $K^{\triangleright } \star A \rightarrow \operatorname{\mathcal{E}}$ can be extended to a morphism $K^{\triangleright } \star B \rightarrow \operatorname{\mathcal{E}}$, so that condition $(\ast _ A)$ is satisfied if and only if condition $(\ast _ B)$ is satisfied. Consequently, to show that every object of $\operatorname{Fun}'_{/\operatorname{\mathcal{E}}}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is a $V$-colimit diagram, it suffices to verify condition $(\ast _{A})$ in the special case where $A$ is an $\infty$-category (see Corollary 4.1.3.3).

We will deduce Theorem 7.3.7.1 from the following special case:

Proposition 7.3.7.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable inner fibration of $\infty$-categories, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $V': \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ denote the functor given by postcomposition with $V$. Let $\operatorname{\mathcal{K}}$ be an $\infty$-category and let $f: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ be a functor, corresponding to a morphism $\operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{B}}$ and a functor $F: \operatorname{\mathcal{K}}^{\triangleleft } \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. If $F$ is $V$-left Kan extended from $\operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, then $f$ is a $V'$-colimit diagram.

Proof. Without loss of generality, we may assume that $V$ is an isofibration, so that $V'$ is also an isofibration (Proposition 4.5.9.18). Fix a morphism $\overline{f}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$, and let

$\operatorname{Fun}'_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) ) \subseteq \operatorname{Fun}_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) )$

denote the full subcategory spanned by those morphisms $\operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ which correspond to functors $\operatorname{\mathcal{K}}^{\triangleleft } \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which are $V$-left Kan extended from $\operatorname{\mathcal{K}}$, and let $\operatorname{Fun}'_{ / \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}},\operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) )$ denote its essential image under the restriction map

$\operatorname{Fun}_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) ).$

We will complete the proof by showing that every object of $\operatorname{Fun}'_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) )$ is a $V'$-colimit diagram in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. By virtue of Remark 7.3.7.12, it will suffice to verify condition $(\ast _{\operatorname{\mathcal{A}}})$ of Lemma 7.3.7.10 for every $\infty$-category $\operatorname{\mathcal{A}}$.

Fix a morphism $\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ extending $\overline{f}$, which we identify with a diagram $\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ and a functor

$\overline{G}: (\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}.$

Let $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( ( \operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote denote the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( (\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ given by the inverse image of $\operatorname{Fun}'_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) )$, and define $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( (\operatorname{\mathcal{K}}\star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ similarly. We wish to show that the restriction map

$\theta : \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( ( \operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( ( \operatorname{\mathcal{K}}\star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$

is an equivalence of $\infty$-categories. Unwinding the definitions (and using the existence criterion of Proposition 7.3.5.5), we see that a functor $G \in \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( (\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ belongs to the subcategory $\operatorname{Fun}'_{ / \operatorname{\mathcal{E}}}( ( \operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ if and only if it is $V$-left Kan extended from $( \operatorname{\mathcal{K}}\star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, and that a functor $G_0 \in \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( (\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ belongs to $\operatorname{Fun}'_{ / \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) }( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) )$ if and only if admits a left Kan extension $G: (\operatorname{\mathcal{K}}^{\triangleright } \star \operatorname{\mathcal{A}}) \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $V \circ G = \overline{G}$. Applying Theorem 7.3.6.14, we conclude that $\theta$ is a trivial Kan fibration. $\square$

Example 7.3.7.14. In the situation of Proposition 7.3.7.13, suppose that $\operatorname{\mathcal{B}}= \Delta ^0$, so that we can identify $F$ with a functor from $\operatorname{\mathcal{K}}^{\triangleleft } \times \operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. Let $X$ denote the cone point of $\operatorname{\mathcal{K}}^{\triangleright }$. For each object $C \in \operatorname{\mathcal{C}}$, the inclusion map $\operatorname{\mathcal{K}}\times \{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{K}}\times \operatorname{\mathcal{C}}_{/C}$ is right cofinal (see Corollary 7.2.1.19). Applying Corollary 7.2.2.2, we deduce that $F$ is $V$-left Kan extended from $\operatorname{\mathcal{K}}\times \operatorname{\mathcal{C}}$ if and only if the induced map

$\operatorname{\mathcal{K}}^{\triangleleft } \simeq \operatorname{\mathcal{K}}^{\triangleleft } \times \{ C \} \hookrightarrow \operatorname{\mathcal{K}}^{\triangleleft } \times \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a $V$-colimit diagram for each $C \in \operatorname{\mathcal{C}}$. Proposition 7.3.7.13 asserts that, if this condition is satisfied, then $F$ determines a $V'$-colimit diagram $\operatorname{\mathcal{K}}^{\triangleleft } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$; this is a special case of Corollary 7.1.6.11.

Proof of Theorem 7.3.7.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an inner fibration of $\infty$-categories, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $V': \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ be the functor given by postcomposition with $V'$. Suppose we are given a functor $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ and a full subcategory $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$. Assume that the induced map $F: \operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$; we ish show that $f$ is $V'$-left Kan extended from $\operatorname{\mathcal{A}}^{0}$. Fix an object $A \in \operatorname{\mathcal{A}}$ and set $\operatorname{\mathcal{A}}^{0}_{/A} = \operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{A}}} \operatorname{\mathcal{A}}_{/A}$; we wish to show that the composite map

$(\operatorname{\mathcal{A}}^{0}_{/A})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{A}}_{/A})^{\triangleright } \rightarrow \operatorname{\mathcal{A}}\xrightarrow {f} \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}},\operatorname{\mathcal{D}})$

is a $V$-colimit diagram. Let $F_{A}$ denote the composition

$( \operatorname{\mathcal{A}}^{0}_{/A})^{\triangleright } \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}.$

By virtue of Proposition 7.3.7.13, it will suffice to show that $F_{A}$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0}_{/A} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$. Let $X$ denote the cone point of $(\operatorname{\mathcal{A}}^{0}_{/A})^{\triangleright }$, let $B$ denote its image in $\operatorname{\mathcal{B}}$, and let $C \in \operatorname{\mathcal{C}}$ be an object satisfying $U(C) = B$. Unwinding the definitions, we see that $F_{A}$ is $V$-left Kan extended from $\operatorname{\mathcal{A}}^{0}_{/A} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ at the object $(X,C)$ if and only if the diagram

$(\operatorname{\mathcal{A}}^{0}_{/A} \times _{\operatorname{\mathcal{B}}_{/B} } \operatorname{\mathcal{C}}_{/C} )^{\triangleright } \rightarrow ( \operatorname{\mathcal{A}}^{0}_{/A} )^{\triangleright } \times _{ (\operatorname{\mathcal{B}}_{/B})^{\triangleright } } (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{A}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a $V$-colimit diagram. This follows from our assumption that $F$ is $V$-left Kan extended from the full subcategory $\operatorname{\mathcal{A}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ at the object $(A,C)$. $\square$