Proposition 4.5.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{\simeq }$ denote its core (Construction 4.4.3.1). For every Kan complex $X$, composition with the inclusion map $\iota : \operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces a bijection
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( X, \operatorname{\mathcal{C}}^{\simeq } ) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( X, \operatorname{\mathcal{C}}). \]