Remark 4.7.1.2. In the formulation of Definition 4.7.1.1, we can replace the morphism space $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ by any Kan complex which is homotopy equivalent to $M$. For example, we can replace $M$ by the left pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(C,X)$ or the right pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(C,X)$ (see Proposition 4.6.5.10).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$