Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 5.4.3.7. The coslice comparison functor $F: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ induces an isomorphism of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{Kan}}_{\ast } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{S}}}_{\ast }$, where $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ denotes the homotopy category of pointed Kan complexes (Construction 3.2.1.10).

Proof. It follows from Propositions 2.4.6.8 and 5.4.3.5 that the functor $\mathrm{h} \mathit{F}$ is an equivalence of categories. Since it is bijective on objects, it is an isomorphism of categories. $\square$