Example 6.2.2.3. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1) and let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between small $\infty $-categories, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{QC}}$. Then:
The functor $F$ is a $\operatorname{\mathcal{S}}$-local equivalence (in the sense of Definition 6.2.2.1) if and only if it is a weak homotopy equivalence of simplicial sets.
The functor $F$ is a $\operatorname{\mathcal{S}}$-colocal equivalence (in the sense of Warning 6.2.2.2) if and only if the map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence.