Kerodon

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Corollary 9.4.5.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a categorical equivalence if and only if it is a Morita equivalence and the induced map of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective.

Proof. Using Remark 9.4.5.5 (and Proposition 4.1.3.2), we can reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, in which case the desired result follows from the criterion of Proposition 9.4.5.8. $\square$