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Remark Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category with underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$. Then the simplicial functor $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ induces a functor of ordinary categories $u_0: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which can be described as follows:

  • On objects, the functor $u_0$ is the identity map from $\operatorname{Ob}( \operatorname{\mathcal{C}}) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ to itself.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is a surjection, which we will denote by $f \mapsto [f]$.

  • Given a pair of morphisms $f,g; X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ having the same source and target, we have $[f] = [g]$ if and only if $f$ and $g$ belong to the same connected component of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.