Warning 2.4.0.1. The ordinary nerve functor $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding from the category $\operatorname{Cat}$ of small categories to the category $\operatorname{Set_{\Delta }}$ of simplicial sets (Proposition 1.3.3.1). However, the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ is not fully faithful when regarded as a functor of ordinary categories. Phrased differently, the adjoint functors
associate to each simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ a counit map $v: \operatorname{Path}[ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}_{\bullet })]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$, which is almost never an isomorphism of simplicial categories. However, we will see later that $v$ is a weak equivalence of simplicial categories whenever $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan (
). Moreover, the construction $\operatorname{\mathcal{C}}_{\bullet } \mapsto \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ establishes an equivalence from the homotopy theory of (locally Kan) simplicial categories $\operatorname{\mathcal{C}}_{\bullet }$ with the homotopy theory of $\infty $-categories ().