On the Cocartesian Image of Preorders and Equivalence Relations in Regular Categories

In a regular category $$\mathbb {E}$$ , the direct image along epimorphism f of preorder is not in general. Set, its best approximation then cocartesian above f. category, existence such S actually equivalent to supremum $$R[f]\vee S$$ among preorders. We investigate here some conditions ensuring these images or equivalently suprema. They apply two very dissimilar contexts: any topos with suprema countable chains subobjects n-permutable category.