Variant 7.3.1.5 (Left Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose we are given a simplicial set $K$ together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F \circ \delta $, as indicated in the diagram
We will say that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the following condition is satisfied:
- $(\ast _ C)$
Let $\beta _{C}$ denote a composition of the natural transformations
\[ F_0|_{ K_{/C} } \xrightarrow {\beta } (F \circ \delta )|_{ K_{/C} } \xrightarrow { F(\gamma ) } \underline{ F(C) } \](formed in the $\infty $-category $\operatorname{Fun}( K_{/C}, \operatorname{\mathcal{D}})$), where $\gamma : \delta |_{ K_{/C} } \rightarrow \underline{C}$ is defined in Notation 7.3.1.1. Then $\beta _{C}$ exhibits $F(C)$ as a colimit of the diagram
\[ K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]in the sense of Definition 7.1.1.1.