How to call such a morphism?

Question

I'm bumping into a property that I would like the morphisms in my own favourite category to have, and I would like to know if it already has a name.

Suppose we have a morphism $r : X \rightarrow Y$ such that for every $s : S \rightarrow X$ there exists an $h : Y \rightarrow S$ (not necessarily unique) with $r \cdot s \cdot h = 1_Y$. How would you call such a morphism? Is there a name for it in literature?

Note 1: obviously, r is a retract, but that's not all is it?

Note 2: The property also turns out to make sense in a concrete category. In such a category, how would you call a function $r : |X| \rightarrow Y$ such that for every morphism $s : S \rightarrow X$ there is a function $h : Y \rightarrow |S|$ with $r \cdot s \cdot h = 1_Y$? (I write $|X|$ here to denote the application of the forgetful functor to Set.)

Thanks, Pieter

Answer 1

Maybe this property is not so useful afterall. For $r$ to have the above property, $r \cdot s$ needs to be a retract for any given $s$. So, having this property at least means that every morphism into X is epic...