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Corollary 7.4.6.9. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or $G$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then 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.

Proof. Applying Proposition 7.4.6.7 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, we deduce that $\theta $ is a homotopy equivalence. Since the restriction map $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$ is an inner fibration of $\infty $-categories (Corollary 4.1.4.2), the map $\theta $ is also a Kan fibration (Proposition 4.6.1.19), and therefore a trivial Kan fibration (Proposition 3.3.7.4). $\square$