Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras

Let (R∨,R) be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra H with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld modules over the bosonizations of R and of R∨, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semi-simple Yetter-Drinfeld modules over H.