Corollary 4.8.5.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 1$. Then $F$ is $n$-full if and only if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of morphism spaces
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is $(n-1)$-full.