Kerodon

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Proposition 4.4.3.16. 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 $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}})$.

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 1.4.1.6, since $u$ is automatically an isomorphism in the $\infty $-category $X$ (Proposition 1.3.6.10). $\square$