Proposition 4.4.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a Kan complex. Then composition with the inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $F: X \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. To show that $F$ factors through the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$, we must show that for every edge $u: x \rightarrow y$ of the Kan complex $X$, the image $F(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$. This follows from Remark 1.5.1.6, since $u$ is automatically an isomorphism in the $\infty $-category $X$ (Proposition 1.4.6.10). $\square$