Proposition 7.3.5.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose we are given diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$. Then:
The diagram $F_0$ admits a left Kan extension along $\delta $ if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}} \]
has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.
The diagram $F_0$ admits a right Kan extension along $\delta $ if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ K_{C/} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}} \]
has a limit in the $\infty $-category $\operatorname{\mathcal{D}}$.
Proof of Proposition 7.3.5.1.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose we are given diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the composite map
\[ K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}} \]
has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$. We wish to show that $F_0$ has a left Kan extension along $\delta $ (the converse assertion is immediate from the definitions, and the analogous assertion for right Kan extensions will follow by a similar argument). Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\iota : K \hookrightarrow \operatorname{\mathcal{K}}$, 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 $\overline{\delta }: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ and $\overline{F}_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, respectively (Proposition 4.1.3.1). For every object $C \in \operatorname{\mathcal{C}}$, the induced map $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \hookrightarrow \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is a categorical equivalence (Corollary 5.6.7.6), and therefore right cofinal (Corollary 7.2.1.13). Applying Proposition 7.2.2.9, we deduce that the composite map
\[ \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{K}}\xrightarrow { \overline{F}_0 } \operatorname{\mathcal{D}} \]
has a colimit in $\operatorname{\mathcal{D}}$. Corollary 7.3.5.8 now guarantees that the functor $\overline{F}_0$ admits a left Kan extension $\overline{F}: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Set $F = \overline{F}|_{\operatorname{\mathcal{C}}}$. Applying Proposition 7.3.2.11, we obtain a natural transformation $\overline{\beta }: \overline{F}_0 \rightarrow F \circ \overline{\delta }$ which exhibits $F$ as a left Kan extension of $\overline{F}_0$ along $\overline{\delta }$. Since $\iota $ is a categorical equivalence, it follows that $\overline{\beta }$ restricts to a natural transformation $F_0 \rightarrow F \circ \delta $ which exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ (Proposition 7.3.1.15).
$\square$