Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Theorem 1.4.3.7. Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Fun}( S_{\bullet }, \operatorname{\mathcal{D}})$ is an $\infty $-category.

Proof of Theorem 1.4.3.7. Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. We wish to show that the simplicial set $\operatorname{Fun}( S_{\bullet }, \operatorname{\mathcal{D}})$ is an $\infty $-category. By virtue of Theorem 1.4.6.1, it will suffice to show that the restriction map

\[ r: \operatorname{Fun}( \Delta ^2, \operatorname{Fun}( S_{\bullet } , \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, \operatorname{Fun}(S_{\bullet }, \operatorname{\mathcal{D}}) ) \]

is a trivial Kan fibration. Note that we can identify $r$ with the canonical map

\[ \operatorname{Fun}( S_{\bullet }, \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( S_{\bullet }, \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{D}}) ), \]

which is a trivial Kan fibration by virtue of Corollary 1.4.5.6 and Theorem 1.4.6.1. $\square$