Definition 7.3.1.2 (Right 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 $\alpha : F \circ \delta \rightarrow F_0$, as indicated in the diagram
We will say that $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the following condition is satisfied:
- $(\ast _ C)$
Let $\alpha _{C}$ denote a composition of the natural transformations
\[ \underline{ F(C) } \xrightarrow { F( \gamma ' ) } (F \circ \delta )|_{ K_{C/} } \xrightarrow { \alpha } F_0|_{ K_{C/} } \](formed in the $\infty $-category $\operatorname{Fun}( K_{C/}, \operatorname{\mathcal{D}})$), where $\gamma ': \underline{C} \rightarrow \delta |_{ K_{C/} }$ is defined in Notation 7.3.1.1. Then $\alpha _{C}$ exhibits $F(C)$ as a limit of the diagram
\[ K_{C/} = \operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]in the sense of Definition 7.1.1.1.