Kerodon

Example 5.4.8.3. Let $\operatorname{\mathcal{C}}$ be a simplicial category. Suppose that, for every pair of objects $X$ and $Y$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the inclusion $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ of Remark 2.4.3.8 carries every $2$-simplex of the ordinary nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to a thin $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$.