Corollary 7.3.8.8. Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be full subcategories. Then $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it satisfies the following pair of conditions:
- $(1)$
The functor $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}$.
- $(2)$
The restriction $\overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.