Yetter-Drinfeld categories for quasi-Hopf algebras

We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra $H$ are isomorphic. We prove also that the category ${\text{yd}}^{\mathrm{f}\mathrm{d}}$ of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute explicitly the canonical isomorphisms in ${\text{yd}}^{\mathrm{f}\mathrm{d}}$. Finally, we show that certain duals of $H_0$, the braided Hopf algebra introduced in \cite{bn,bpv}, are isomorphic as braided Hopf algebras if $H$ is a finite dimensional triangular quasi-Hopf algebra.