$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.6.9. 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.
Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Assume either that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or that $G$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$. Applying Proposition 7.3.6.7 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, we deduce that 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. 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 4.6.1.21), and therefore a trivial Kan fibration (Proposition 3.3.7.6).
$\square$