Remark 4.6.8.18. It follows from Corollary 4.6.8.17 that the left-pinched morphism space $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$ depends only on the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, and not on any other features of the simplicial category $\operatorname{\mathcal{C}}$. In particular, there is a canonical isomorphism
\[ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^0, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ). \]