Equivalence between categories of supermodules

Let R be commutative algebra with unit element. Given any R-module U, there exists a category CU whose objects are all pairs (N,M) where N is direct sum of U and M contains U. It has been shown that if W is a direct summand of U then there exists a functor from CU to CW. Further, if W and U are isomorphic then there exists category equivalence between CU and C W.