  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 applied to two very
dissimilar contexts: any topos $\mathbb E$ with suprema of chains of subobjects
or any $n$-permutable regular category.
