Example 7.3.1.7. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be a diagram. Let $\delta : K \rightarrow \Delta ^0$ be the projection map and let $F: \Delta ^0 \rightarrow \operatorname{\mathcal{D}}$ be the functor corresponding to an object $Y \in \operatorname{\mathcal{D}}$. Then:
A natural transformation $\alpha : \underline{Y} = (F \circ \delta ) \rightarrow F_0$ exhibits $Y$ as a limit of $F_0$ (in the sense of Definition 7.1.1.1) if and only if it exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ (in the sense of Definition 7.3.1.2).
A natural transformation $\beta : F_0 \rightarrow (F \circ \delta ) = \underline{Y}$ exhibits $Y$ as a colimit of $F_0$ (in the sense of Definition 7.1.1.1) if and only if it exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ (in the sense of Variant 7.3.1.5).