In this paper we introduce and investigate M-cofaithful modules. A module N 2 σ[M] is called M-cofaithful if for every o 6= f 2 HomR(N; X) with X 2 σ[M], HomR(X; M)f 6= 0. We show that if N is an M-cofaithful weak supplemented module and HomR(N; M) a noetherian S-module, then there exists an order-preserving correspondence between the cocolsed R-submodules of N and the closed S-submodules of HomR(N; M), where S = EndR(M). Some applications are: (1) the connection between M ;s being a lifting module and EndR(M);s being an extending ring; (2) the equality between the hollow dimension of a quasi-injective coretractable module M and the uniform dimension of EndR(M)
