Corollary 7.3.4.8. 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:
- $(1)$
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 $F|_{\operatorname{\mathcal{K}}} = F_0$.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ \operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}} \]admits a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.