Call a subobject normal if it's an equivalence class of some internal equivalence relation on the codomain. In a finitely complete category, normality is closed under finite intersection and finite product and is also pullback stable. This can be proved using a metatheorem which reduces to verification in the category of sets.
In the category of groups, normality is also closed under joins in the subobject poset (see this MSE question). Is the finite analogue of this state true in some nice protomodular categories, e.g homological ones?