Corollary 7.3.4.2. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{K}}$.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the functor
\[ F_{C}: \operatorname{\mathcal{K}}_{C}^{\triangleright } \simeq \operatorname{\mathcal{K}}_{C} \star _{ \{ C\} } \{ C\} \hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]is a colimit diagram.