  We investigate adjoint and Frobenius pairs 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,
which leads to a reasonable notion of Frobenius coring extension. When applied
to corings stemming from entwining structures, we obtain new results in this
setting and in graded ring theory.
