Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.5.2.14. For every simplicial category $\operatorname{\mathcal{C}}$, let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$. Then there are canonical isomorphisms of simplicial sets

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\triangleleft } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\triangleleft } \quad \quad \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\triangleright } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\triangleright }, \]

which are uniquely determined by the requirements that they restrict to the identity on $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and preserve the cone points. In other words, the formation of cones is compatible with the homotopy coherent nerve.