The goal of this work is to develop a functional transfer of properties between a module A and the category *M *E of right modules over its endomorphism ring *E that is more sensitive than the traditional starting point Hom (A, .). The main result is a factorization q * A t* of the left adjoint T *A of Hom(, .), where t* is a category equivalence and q* is a forgetful functor. Applications include a characterization of the finitely generated submodules...