Definition 4.8.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. We will say that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-faithful 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 $(n-2)$-truncated (Definition 3.5.9.1).