Theorem 1.5.3.7. Let $S$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Fun}( S, \operatorname{\mathcal{D}})$ is an $\infty $-category.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Theorem 1.5.3.7. Let $S$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. We wish to show that the simplicial set $\operatorname{Fun}( S, \operatorname{\mathcal{D}})$ is an $\infty $-category. By virtue of Theorem 1.5.6.1, it will suffice to show that the restriction map
\[ r: \operatorname{Fun}( \Delta ^2, \operatorname{Fun}( S , \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, \operatorname{Fun}(S, \operatorname{\mathcal{D}}) ) \]
is a trivial Kan fibration. Note that we can identify $r$ with the canonical map
\[ \operatorname{Fun}( S, \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( S, \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{D}}) ), \]
which is a trivial Kan fibration by virtue of Corollary 1.5.5.7 and Theorem 1.5.6.1. $\square$