Corollary 4.6.8.17. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be a simplicial category containing objects $X$ and $Y$. Then we have a canonical bijection
\[ \{ \textnormal{Morphisms $K \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$} \} \simeq \{ \textnormal{Morphisms $\Phi (K) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$} \} . \]