Remark 4.8.7.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is categorically $n$-connective if and only if it satisfies the following pair of conditions:
The functor $F$ is locally $(n-1)$-connective. That is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]is $(n-1)$-connective.
If $n \geq 0$, then $F$ is essentially surjective.
See Corollary 4.8.5.22.