  Starting from a comonad G on a category A, and a functor L : B -> A with a
right adjoint R : A -> B, we will give a parametrization of the functors K from
B to the category of all G-coalgebras that factorize throughout L in terms of
homomorphisms of comonads from LR to G. Next, we will see under which
conditions one of these functors K admits a right adjoint D. We will
characterize when D is full and faithful, and we will conclude our general
results by characterizing when K establishes an equivalence between the
category B and the category of G-coalgebras. Obviously, the functors
characterized in this way are, a fortiori, comonadic but, in contrast with the
approach of Beck's Theorem, the comonad G is here given beforehand, and each
functor K corresponds to a ``representation'' of G. Beck's theorem deals with
the situation where G = LR. We apply our general results to the case of corings
over firm rings.

Category Theory;Rings and Algebras