faithful functor

Definition

Given categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ is called faithful if for all $X,Y\in \mathcal{C}$, the maps

$$F_{X,Y}:{\text{Hom}}_{\mathcal{C}}(X,Y)⟶{\text{Hom}}_{\mathcal{D}}(F(X),F(Y))$$

is an injection.