We can also prove that Yetter-Drinfeld condition implies that $\Psi$ is a braiding as follows.

Let $u, v \in V, h \in H$. Then

\begin{align} & \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes h) \\ & = \Psi_{12} \Psi_{23}( (u_{(-1)}.v) \otimes u_{(0)} \otimes h ) \\ & = ( u_{(-1)}.v )_{(-1)}( ( (u_{(0)})_{(-1)} ).h ) \otimes (u_{(-1)}.v)_{(0)} \otimes ( u_{(0)} )_{(0)} \\ & = ( (u_{(-1)})_{(1)}.v )_{(-1)}( ( (u_{(-1)})_{(2)} ).h ) \otimes ((u_{(-1)})_{(1)}.v)_{(0)} \otimes u_{(0)} (1) \\ & \Psi_{23} \Psi_{12} \Psi_{23} (u \otimes v \otimes h) \\ & = (u_{(-1)} v_{(-1)}).h \otimes (u_{(0)})_{(-1)} \otimes ( u_{(0)} )_{(0)} \\ & = ( (u_{(-1)})_{(1)} v_{(-1)}).h \otimes (u_{(-1)})_{(2)} \otimes u_{(0)}. (2) \end{align}

Therefore the Yetter-Drinfeld condition implies that (1)=(2).

We can also prove that Yetter-Drinfeld condition implies that $\Psi$ is a braiding as follows.

Let $u, v, w \in V $. Then

\begin{align} & \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes w) \\ & = \Psi_{12} \Psi_{23}( (u_{(-1)}.v) \otimes u_{(0)} \otimes w ) \\ & = \Psi_{12} ( (u_{(-1)}.v) \otimes (u_{(0)})_{(-1)}.w \otimes (u_{(0)})_{(0)} ) \\ & = ( u_{(-1)}.v )_{(-1)}.( ( (u_{(0)})_{(-1)} ).w ) \otimes (u_{(-1)}.v)_{(0)} \otimes ( u_{(0)} )_{(0)} \\ & = ( (u_{(-1)})_{(1)}.v )_{(-1)}.( (u_{(-1)})_{(2)}.w ) \otimes ((u_{(-1)})_{(1)}.v)_{(0)} \otimes u_{(0)} \\ & = ( ((u_{(-1)})_{(1)}.v )_{(-1)} (u_{(-1)})_{(2)} ).w \otimes ((u_{(-1)})_{(1)}.v)_{(0)} \otimes u_{(0)}, \end{align} \begin{align} & \Psi_{23} \Psi_{12} \Psi_{23} (u \otimes v \otimes w) \\ & = \Psi_{23} \Psi_{12} (u \otimes v_{(-1)}.w \otimes v_{(0)}) \\ & = \Psi_{23} ((u_{(-1)}v_{(-1)}).w \otimes u_{(0)} \otimes v_{(0)}) \\ & = (u_{(-1)} v_{(-1)}).w \otimes (u_{(0)})_{(-1)}.v_{(0)} \otimes ( u_{(0)} )_{(0)} \\ & = ( (u_{(-1)})_{(1)} v_{(-1)}).w \otimes (u_{(-1)})_{(2)}.v_{(0)} \otimes u_{(0)}. \end{align}

Therefore the Yetter-Drinfeld condition \begin{align} & (h_{(1)}.v )_{(-1)} h_{(2)} \otimes (h_{(1)}.v)_{(0)}\\ & = h_{(1)} v_{(-1)} \otimes h_{(2)}.v_{(0)}. \end{align} implies that (every $u_{(-1)}$ is some $h \in H$) \begin{align} & ((u_{(-1)})_{(1)}.v )_{(-1)} (u_{(-1)})_{(2)} \otimes ((u_{(-1)})_{(1)}.v)_{(0)} \otimes u_{(0)} \\ & = (u_{(-1)})_{(1)} v_{(-1)} \otimes (u_{(-1)})_{(2)}.v_{(0)} \otimes u_{(0)}. \end{align} Therefore \begin{align} & \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes w) = \Psi_{23} \Psi_{12} \Psi_{23} (u \otimes v \otimes w). \end{align}