I didn’t say I wouldn’t want to define this term. The question was just what term to use. I didn’t want to invent a new term if there was already some term in use.

That’s interesting. Thanks for the hint. However, wouldn’t $\mathcal{R}$ need to be an equivalence relation in order to be a congruence? In my use case, $\mathcal{R}$ typically isn’t symmetric and might not even be reflexive or transitive.

Thanks for the pointer. In fact, my question could be rephrased as “Is every monotonic endofunctor on $\mathbf{Rel}$ a relator?”. The fact that the authors explicitly require preservation of conversion suggests that this property does not follow from the other ones. However, it would be great to have an example of a monotonic endofunctor on $\mathbf{Rel}$ that is not a relator (that is, doesn’t preserve conversion).