Kerodon

Corollary 5.4.3.10. The coslice comparison functor $\Phi : \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ induces an isomorphism of homotopy categories $\mathrm{h} \mathit{\Phi }: \mathrm{h} \mathit{\operatorname{Kan}}_{\ast } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{S}}}_{\ast }$, where $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ denotes the homotopy category of pointed Kan complexes (Construction 3.2.1.10).