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Let A be a ring and M_A the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor −⊗_AB:M_A→M_A is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor −⊗_AC:M_A→M_A is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗_AB and comodules (or coalgebras) of −⊗_AC are well studied in the literature. On the other hand, the right adjoint endofunctors Hom_A(B,−) and Hom_A(C,−) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of Hom_A(B,−)-comodules is isomorphic to the category of B-modules, while the category of Hom_A(C,−)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their …