Remark 9.3.1.2. In the formulation of Definition 9.3.1.1, we can replace $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ by any Kan complex which is homotopy equivalent $M$. For example, we can replace $M$ by the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(Y,X)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,X)$ (see Proposition 4.6.5.10).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$