Kerodon

Corollary 7.3.8.17. Let $\overline{\operatorname{\mathcal{C}}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be a full subcategory, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $F$ admits a left Kan extension $\overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ if and only if the restriction $F|_{\operatorname{\mathcal{C}}^{0}}$ admits a left Kan extension $\overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$.