  Let $(R^{\vee},R)$ 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 $R^{\vee}$,
respectively. As an application of this very general category isomorphism we
obtain a natural proof of the existence of reflections of Nichols algebras of
semisimple Yetter-Drinfeld modules over $H$.
  Key words: Hopf algebras, quantum groups, Weyl groupoid
