Remark 4.8.6.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq -1$. Then $F$ is essentially $n$-categorical if and only if, 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 $(n-1)$-truncated. This follows by combining Example 4.8.6.3 with Corollary 4.8.5.22.