finitely cocomplete category of "finite injections" ... ??as "walking epi" ??

finitely cocomplete category of "finite maps" .... ???as "walking map" ... ??...

???theory interpretations between these ???....

"epi as special case of map" ... ???... ???mapping a finite map to its image inclusion here (...) as preserving finite colimits ... ??but re-including the finite injections among the finite maps as _not_ preserving finite colimits?? ??something about cokernel where "two codomain elements are identified together " ... ??as not preserved... ???because of whether domain elements going to them are also identified together... ???or something?? ....