$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.3.4.4. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The functor $F_0$ admits a left Kan extension along $\delta $.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the induced diagram
\[ \operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}} \]
has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.
Proof of Proposition 7.3.4.4.
Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ \operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}} \]
has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$. Applying Corollary 7.3.4.8, we deduce that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{K}}$ and satisfies $F|_{\operatorname{\mathcal{K}}} = F_0$. Applying Proposition 7.3.1.15, we see that the restriction $F|_{\operatorname{\mathcal{C}}}$ is a left Kan extension of $F_0$ along $\delta $.
$\square$