Let $\mathcal{C}$ be a bicategory.
A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably faithful1 if, for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the functor
given by postcomposition by $f$ is faithful.
In detail, $f$ is representably faithful if, for all diagrams in $\mathcal{C}$ of the form
then $\alpha =\beta $.
Here are some examples of representably faithful morphisms.
Representably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the faithful functors; see Chapter 11: Categories, Item 2 of Proposition 11.6.1.1.2.
Representably Faithful Morphisms in $\boldsymbol {\mathsf{Rel}}$. Every morphism of $\boldsymbol {\mathsf{Rel}}$ is representably faithful; see Chapter 8: Relations, 
of .
