Let $\mathcal{C}$ be a bicategory.
A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably faithful if, for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the functor
given by precomposition by $f$ is faithful.
In detail, $f$ is corepresentably faithful if, for all diagrams in $\mathcal{C}$ of the form
then $\alpha =\beta $.
Here are some examples of corepresentably faithful morphisms.
Corepresentably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The corepresentably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 11: Categories, Item 5 of Proposition 11.6.1.1.2.
Corepresentably Faithful Morphisms in $\boldsymbol {\mathsf{Rel}}$. Every morphism of $\boldsymbol {\mathsf{Rel}}$ is corepresentably faithful; see Chapter 8: Relations, Item 1 of Proposition 8.4.10.1.1.
