Proposition 7.3.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $F_0$ denote the restriction of $F$ to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, and let $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ denote the inclusion functor. Then:
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}: F_0 \rightarrow F \circ \iota $ exhibits $F$ as a left Kan extension of $F_0$ along $\iota $ (in the sense of Variant 7.3.1.5).
The functor $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}_{F_0}: F \circ \iota \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\iota $ (in the sense of Definition 7.3.1.2).