  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.
