Corollary 7.3.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^0 }$ be a natural transformation of functors from $\operatorname{\mathcal{C}}^{0}$ to $\operatorname{\mathcal{D}}$. Then $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ (in the sense of Variant 7.3.1.5) if and only if the following pair of conditions is satisfied:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1).
- $(2)$
The natural transformation $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$.