Kerodon

Warning 5.5.2.7. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$ (Construction 2.4.6.1). For every object $X \in \operatorname{\mathcal{C}}$, there is a natural comparison map $\mathrm{h} \mathit{(\operatorname{\mathcal{C}}_{/X})} \rightarrow (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$, which carries an object $(C,f)$ of the slice simplicial category $\operatorname{\mathcal{C}}_{/X}$ to the object $(C, [f] )$ of the slice category $(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$, where $[f] \in \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } )$ denotes the homotopy class of $f$. Beware that this functor is generally not an equivalence of categories (see Warning 3.2.1.11).