Kerodon

Example 4.5.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is a categorical equivalence (in the sense of Definition 4.5.3.1) if and only if it is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10). Both conditions are equivalent to the assertion that for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection $ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.