Let A$A$ be a (finite-dimensional graded cocommutative) Hopf algebra over a field k$k$, E$E$ be a Hopf subalgebra, and R=A \otimes_E k$R=A \otimes_E k$. Then the comultiplication on A$A$ induces a coalgebra structure on R$R$. Furthermore, R$R$ is a coalgebra in the monoidal category of A$A$-modules, with A$A$ acting on R \otimes R$R \otimes R$ diagonally via the comultiplication. Define an internal R$R$-comodule to be an object M$M$ which is simultaneously an A$A$-module and an R$R$-comodule such that the structure map M \to R \otimes M$M \to R \otimes M$ is a map of A$A$-modules, for the diagonal A$A$-module structure on the tensor product.
A$A$ itself is naturally an internal R$R$-comodule, via the comultiplication A \to A \otimes A \to R \otimes A$A \to A \otimes A \to R \otimes A$. For any E$E$-module N$N$, A \otimes_E N$A \otimes_E N$ then inherits an internal R$R$-comodule structure from A$A$. Conversely, if M$M$ is an internal R$R$-comodule, N={m:d(m)=1 \otimes m}$N={m:d(m)=1 \otimes m}$ is an E$E$-module, where d:M \to R \otimes M$d:M \to R \otimes M$ is the structure map.
Is it true (possibly under some reasonable niceness hypotheses) that these two functors between E-modules and internal R$R$-comodules are inverse? In particular, I'd like to interpret this in terms of faithfully flat descent: A$A$ is faithfully flat over E$E$, and I want to say that for an A$A$-module M$M$, there is a natural bijection between descent data that allows us to identify M=A \otimes_E N$M=A \otimes_E N$ for an E$E$-module N$N$ and internal R$R$-comodule structures M \to R \otimes M$M \to R \otimes M$.
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.
