direct but long proof can be given: see [@Nenciu Prop. 2.7]. The monoidal structure can be computed using [(\[eq:center2\])]{}. Take $M,N\in {}_H^H{\mathcal{YD}}$, the $H$-coaction on $M\ot_tN$ is given by the formula $$\lambda(1_{(1)}m\ot 1_{(2)}n)=
((\sigma_{M,H}\ot N)\circ (M\ot \sigma_{N,H}))
(1_{(1')}(1_{(1)}m\ot 1_{(2)}n)\ot 1_{(2')}).$$ Observe that $$\begin{aligned}
x&=& 1_{(1')}(1_{(1)}m\ot 1_{(2)}n)\ot 1_{(2')}\\
&=& 1_{(1')}1_{(1)}m\ot 1_{(1'')}1_{(2')}1_{(2)}n\ot 1_{(2'')}=
1_{(1)}m\ot 1_{(1'')}1_{(2)}n\ot 1_{(2'')},\end{aligned}$$ so that $$\begin{aligned}
&&\hspace*{-2cm}
(M\ot \sigma_{N,H})(x)=
1_{(1)}m\ot (1_{(2)}n)_{[-1]}\ot (1_{(2)}n)_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}n_{[-1]}S(1_{(4)})\ot 1_{(3)}n_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}1_{(1')}n_{[-1]}S(1_{(3')})\ot 1_{(2')}n_{[0]}\\
&=& 1_{(1)}m\ot 1_{(2)}n_{[-1]}\ot n_{[0]}\end{aligned}$$ and $${\label{eq:2.5.1}}
\lambda(1_{(1)}m\ot 1_{(2)}n)=m_{[-1]}n_{[-1]}\ot m_{[0]}\ot n_{[0]}.$$ We compute the left $H$-coaction on $H_t$ using [(\[eq:center2bis\])]{} and [(\[eq:2.4.1\])]{}. For any $z\in H_t$, this gives $$\begin{aligned}
\lambda(z)&=& \sigma_{H_t,H}((1_{(1)}\leftact z)\ot 1_{(2)})=
 r_M^{-1}(l_M((1_{(1)}\leftact z)\ot 1_{(2)}))\nonumber\\
&=&
r_M^{-1}(z)= 1_{(1)}z \ot 1_{(2)}=\Delta(z).{\label{eq:2.5.2}}\end{aligned}$$ The braiding and its inverse are given by the formulas $$\sigma_{M,N}(1_{(1)} m\ot 1_{(2)}n)=m_{[-1]}n\ot m_{[0]}~~;~~
\sigma^{-1}_{M,N}(1_{(1)} n\ot 1_{(2)}m)=
m_{[0]}\ot S^{-1}(m_{[-1]})n.$$

A left-right Yetter-Drinfeld module is a $k$-module with a left $H$-action and a right $H$-coaction such that the following conditions hold, for all $m\in M$ and $h\in H$: $$\begin{aligned}
&&\hspace*{-2cm}\rho(m)= m_{[0]}\ot m_{[1]}\in M\ot_t H;{\label{eq:2.6.1}}\\
&&\hspace*{-2cm} h_{(1)}m_{[0]}\ot h_{(2)}m_{[1]}=(h_{(2)}m)_{[0]}\ot 
(h_{(2)}m)_{[1]}h_{(1)}.{\label{eq:2.6.2}}\end{aligned}$$ The category of left-right Yetter-Drinfeld modules and left $H$-linear right $H$-colinear maps is denoted by ${}_H{\mathcal{YD}}^H$.

[\[pr:2.6\]]{} Let $H$ be a weak Hopf algebra with bijective antipode. Then the category ${}_H{\mathcal{YD}}^H$ is 