Simple Situation

Dear categorists, Let C be a category with a distinguished sub-category E and a distinguished family S of morphisms such that for every object x of C there is a unique morphism f_x: x ---> e_x with e_x an object of E so that the following conditions are satisfied: (1) if x is in E then f_x = 1_x (the identity morphism of x), (2) if s: x ---> y is in S then e_y = e_x and f_x = f_y s. Hasn't this simple situation been named and incorporated in some publication on category theory? A reference would be most appreciated. Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]