Corollary 7.3.6.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $F_0 = F|_{\operatorname{\mathcal{C}}^{0}}$ be the restriction of $F$ to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Suppose that the functor $F_0$ admits a left Kan extension to $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
- $(2)$
For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the restriction map
\[ \theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})}( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } ) \]is a homotopy equivalence of Kan complexes.
- $(3)$
For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the restriction map
\[ \theta : \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})}( F|_{\operatorname{\mathcal{C}}^{0} }, G|_{ \operatorname{\mathcal{C}}^{0} } ) \]is a trivial Kan fibration of simplicial sets.