# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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.