# Kerodon

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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.