Given a Setoid/IsEquivalence bundle/structure on type B, and h : A → B, to construct
_≈_ : Rel A _,
by = B._≈_ on h
isRelHomomorphism : IsRelHomomorphism _≈_ B._≈_ h,
by isRelHomomorphism = record { cong = id }
isRelMonomorphism : IsRelMonomorphism _≈_ B._≈_ h,
by isRelMonomorphism = record { isRelHomomorphism = isRelHomomorphism; injective = id }
And... that's just about it!
Followup: extend this to Algebra.Morphism.Construct.On. cf. #2863