A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably faithful1 if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor
\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright ) \]
given by postcomposition by $f$ is faithful.
1Further Terminology: Also called simply a faithful morphism, based on Item 1 of Example 11.1.1.1.3.
