  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.
