Kerodon

Remark 4.5.1.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. If $F$ is an equivalence of $\infty $-categories, then the induced map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence of Kan complexes. This follows from Corollary 4.5.1.6 (and Remark 4.5.1.7): if the isomorphism class $[F]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$, then the homotopy class $[ F^{\simeq } ]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.