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.