Let A be a (finite-dimensional graded cocommutative) Hopf algebra over a field k, E be a Hopf subalgebra, and R=A \otimes_E k. Then the comultiplication on A induces a coalgebra structure on R. Furthermore, R is a coalgebra in the monoidal category of A-modules, with A acting on R \otimes R diagonally via the comultiplication. Define an internal R-comodule to be an object M which is simultaneously an A-module and an R-comodule such that the structure map M \to R \otimes M is a map of A-modules, for the diagonal A-module structure on the tensor product.

A itself is naturally an internal R-comodule, via the comultiplication A \to A \otimes A \to R \otimes A. For any E-module N, A \otimes_E N then inherits an internal R-comodule structure from A. Conversely, if M is an internal R-comodule, N={m:d(m)=1 \otimes m} is an E-module, where d:M \to R \otimes M is the structure map.

Is it true (possibly under some reasonable niceness hypotheses) that these two functors between E-modules and internal R-comodules are inverse? In particular, I'd like to interpret this in terms of faithfully flat descent: A is faithfully flat over E, and I want to say that for an A-module M, there is a natural bijection between descent data that allows us to identify M=A \otimes_E N for an E-module N and internal R-comodule structures M \to R \otimes M.

Sorry if I'm getting some things wrong about what hypotheses are needed for this to make sense; I'm trying to understand this in a specific example and don't know much of the general theory.