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We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.

respectively. The notation ⊗ will stand for the tensor product over k. A category C is said to be k-category if for every M and N in C, Hom C (M, N) is a k-module, and the composition is k-bilinear. An abelian category which is a k-category is said to be k-abelian category. A functor between k-categories is said to be k-functor or k-linear functor if it is k-linear on the k-modules of morphisms. A functor between k-categories is said to be a k-equivalence if it is k-linear and an equivalence.We recall from [25] that an A-coring is an A-bimodule C with two A-bimodule maps ∆ :Coproducts and cokernels (and then inductive limits) in M C exist and they coincide respectively with coproducts and cokernels in the category of right A-modules M A . If A C is flat, then M C is a k-abelian category. Moreover it is a Grothendieck category. When C = A is the obvious A-coring, M A is the category of right A-modules M A . Now assume that the A ′ − A-bimodule M is also a left comodule over an A ′ -coring C ′ with structure map λIn this case, we say that M is a C ′ − C-bicomodule. A morphism of bicomodules is a morphism of right and left comodules. Then we obtain a k-categoryLet Z be a left A-module and f : X → Y a morphism in M A . Following [6, 40.13] we say that f is Z-pure when the functor − ⊗ A Z preserves the kernel of f . If f is Z-pure for every Z ∈ A M then we say simply that f is pure in M A .Let f : M → N be a morphism in C ′ M C , and let ker(f ) be its kernel in A ′ M A . It is easy to show that if ker(f ) is C ′ A ′ -pure and A C-pure, and the followingare injective maps, then ker(f ) is the kernel of f in C ′ M C . This is the case if f is (C ′ ⊗ A ′ C ′ ) A ′pure, A (C ⊗ A C)-pure, and C ′ ⊗ A ′ f is A C-pure (e.g. if C ′ A ′ and A C are flat, or if C is a coseparable A-coring).

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