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Corollary 7.4.6.11. 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 $G: \rightarrow \operatorname{\mathcal{D}}$, 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.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows by applying Corollary 7.4.6.10 in the special case $\operatorname{\mathcal{E}}= \Delta ^{0}$. The equivalence $(2) \Leftrightarrow (3)$ is a special case of Proposition 3.3.7.4, since the morphism $\theta $ is automatically a Kan fibration (see Corollary 4.1.4.2 and Proposition 4.6.1.19). $\square$