Theorem 4.6.8.5. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Then the comparison map
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]
of Construction 4.6.8.3 is a homotopy equivalence of Kan complexes.