Definition 4.8.5.2 (Faithful Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$.