Kerodon

Corollary 7.3.7.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $f: \operatorname{\mathcal{A}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ be a functor of $\infty $-categories, corresponding to a functor $F: \operatorname{\mathcal{A}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Let $\operatorname{\mathcal{A}}^{0} \subseteq \operatorname{\mathcal{A}}$ be a full subcategory. If $F$ is left Kan extended from $\operatorname{\mathcal{A}}^{0} \times \operatorname{\mathcal{C}}$, then $f$ is left Kan extended from $\operatorname{\mathcal{A}}^{0}$.