9.1.1 Representably Faithful Morphisms
Let $\mathcal{C}$ be a bicategory.
A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably faithful 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.
Here are some examples of representably faithful morphisms.
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Representably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the faithful functors; see Chapter 8: Categories, Item 1 of Proposition 8.5.1.1.2.
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Representably Faithful Morphisms in $\textbf{Rel}$. Every morphism of $\textbf{Rel}$ is representably faithful; see Chapter 5: Relations, Item 1 of Proposition 5.3.8.1.1.