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

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