Cometic functors for small concrete categories and an application

Our goal is to derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics. Let A be a small concrete category. We define a functor F, cometic functor, from A to the category Set and a natural transformation \pi, called cometic projection, from F to the inclusion functor of A into Set such that the F-image of every monomorphism A is an injective map and the components of \pi are surjective maps. Also, we give a nontrivial application of F and \pi.