Examples of equivalences of Doi-Koppinen Hopf module categories, including Yetter-Drinfeld modules (Q1280204)

Let \(H\) be a Hopf algebra and \(^H_H{\mathcal M}^H_H\) the category of two-sided two-cosided Hopf modules in the sense of Woronowicz. Then there exists a Doi-Hopf datum \((A,B,D)\) such that the above category is equivalent to the category of Hopf modules in the sense of Doi-Koppinen \(^D_B{\mathcal M}(A)\). This implies a category equivalence between the category of two-sided two-cosided Hopf modules and the category \(^H_H\mathcal{YD}\) of Yetter-Drinfeld (or crossed) modules.