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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 1.4.5.7 and Theorem 1.4.6.1. $\square$