The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)
Is there a name for this construction?
Is there a name for this property of categories?
In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property?