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Remark 5.4.2.13. For every simplicial category $\operatorname{\mathcal{C}}$, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. Then there are canonical isomorphism of categories

\[ \mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\triangleleft } )} \simeq (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\triangleleft } \quad \quad \mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\triangleright } )} \simeq (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\triangleright }. \]

In other words, the formation of cones is compatible with the passage from a simplicial category to its homotopy category.