In any category, monic/epic morphisms are equivalently characterized as the morphisms that map to injective maps under the co-/contravariant Hom-functors. Now I'm interested in the morphisms that map to surjective maps, i.e. that permit extensions/lifts of any morphisms along them. I will call them $\overline{\text{monic}}/\overline{\text{epic}}$ in the following.
I'd like to know if these concepts already have an established name, and if there is some theory around them: Maybe there is some criterion to determine if some morphism is $\overline{\text{monic}}/\overline{\text{epic}}$ analogous to the usual (co)kernel pair criterion for monic/epic morphisms. Are there other types of $\overline{\text{monic}}/\overline{\text{epic}}$ morphisms, as there are effective/regular/strict/strong/extremal mono- and epimorphisms?
These are precisely the split monos and epis respectively. See this blog post.