Proposition 7.3.4.1 (02ZM)

Proposition 7.3.4.1. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Suppose we are given functors of $\infty $-categories $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ and $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F \circ \delta $. The following conditions are equivalent:

$(1)$: The natural transformation $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $.
$(2)$: For each object $C \in \operatorname{\mathcal{C}}$, the restriction of $\beta $ to the fiber $\operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$ determines a natural transformation $F_0|_{ \operatorname{\mathcal{K}}_{C} } \rightarrow \underline{ F(C) }$ which exhibits $F(C)$ as a colimit of the diagram $F_0|_{ \operatorname{\mathcal{K}}_{C} }$ in the $\infty $-category $\operatorname{\mathcal{D}}$.

Proof. By virtue of Corollary 7.2.2.7, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the tautological map

\[ \operatorname{\mathcal{K}}_{C} = \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \{ C\} \hookrightarrow \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \]

is right cofinal. Since $\delta $ is a cocartesian fibration, it will suffice to show that the inclusion map $\{ \operatorname{id}_ C \} \hookrightarrow \operatorname{\mathcal{C}}_{/C}$ is right cofinal (Proposition 7.2.3.12). This follows from Corollary 4.6.7.24, since $\operatorname{id}_{C}$ is a final object of the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22). $\square$