In Definition 2.12 of https://arxiv.org/pdf/1409.3805.pdf, Adamek defines the notion of a cocomplete category having stable monomorphisms, meaning that small coproducts of monomorphisms are monomorphisms, the colimit cocone of any chain of monomorphisms consists of monomorphisms, and the factorizing morphism of any such colimit cocone through any cocone of monomorphisms is itself a monomorphism. Adamek notes that many "set-like" categories and almost all "usual" categories of algebras have stable monomorphisms. I suppose any locally presentable category satisfies the second and third aspects of having stable monomorphisms, but does not necessarily have coproduct-stable monomorphisms.
My question is, is there a more precise/complete characterization of which categories have stable monomorphisms, or more generally stable $\mathcal{M}$-morphisms for a proper factorization system $(\mathcal{E}, \mathcal{M})$?