# Kerodon

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Corollary 5.5.3.10. 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).

Proof. It follows from Propositions 2.4.6.9 and 5.5.3.8 that the functor $\mathrm{h} \mathit{\Phi }$ is an equivalence of categories. Since it is bijective on objects, it is an isomorphism of categories. $\square$