  Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce
all possible categories of Yetter-Drinfeld modules over $H$, and give
explicitly the monoidal and braided structure of them. Then we prove that the
category $^H_H\mathcal{YD}^{fd}$ of finite-dimensional left-left
Yetter-Drinfeld modules is rigid. Finally we will study the braided
cocommunitivity of $H_0$ in $^H_H\mathcal{YD}$.
