Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if \begin{align} ( v \triangleleft h_{(2)} )_{(0)} \otimes h_{(1)} ( v \triangleleft h_{(2)} )_{(1)} = ( v_{(0)} \triangleleft h_{(1)} ) \otimes v_{(1)} h_{(2)}. \quad (1) \end{align}

Let $\Psi: U \otimes W \to W \otimes U$ be a braiding given by \begin{align} \Psi(u \otimes w) = w_{(0)} \otimes ( u \triangleleft w_{(1)} ). \end{align} Then \begin{align} \Psi_{12} \Psi_{23} \Psi_{12} = \Psi_{23} \Psi_{12} \Psi_{23}. \end{align} The algebra $H$ is in the right-right Yetter-Drinfeld category $YD^H_H$. Suppose that $V \in YD^H_H$. Let $\Psi: V \otimes H \to H \otimes V$ be a braiding. Let $u \in H, v \in V, h \in H$. Then \begin{align} & \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes h) \\ & = h_{(0)} \otimes (v_{(0)} \triangleleft h_{(1)}) \otimes ( u \triangleleft v_{(1)} h_{(1)} ) \quad (2) \\ & \Psi_{23} \Psi_{12} \Psi_{23} (u \otimes v \otimes h) \\ & = h_{(0)} \otimes (v \triangleleft h_{(2)})_{(0)} \otimes ( u \triangleleft h_{(1)} ( v \triangleleft h_{(2)} )_{(1)} ). \quad (3) \end{align} How to show that $\Psi$ is a braiding is equavalent to the compatibility condition (1)? My problem is: how to remove $u$ in (2) and (3)? Thank you very much.