Let $\mathcal A$ be a category, $f : A \to B$ be a morphism, $(R_1,R_2 : R \to A)$ an equivalence relation on $A$, $(S_1,S_2 : S \to B)$ an equivalence relation on $B$ and $\operatorname{qt}_{A/R}$, $\operatorname{qt}_{B/S}$ coequalizers, respectively. Let $f$ be "relation preserving", i.e. there exists a (necessarily unique) morphism $\tilde f : R \to S$ making the obvious pair of diagrams commute (see below).
What might be a good notation for the unique morphism $u$ (and perhaps name?) making also the right square in the following diagram commute:
In the set-theoretic case we would of course have $u([x]_R) = [f(x)]_S$ for all $x\in A$.