The category of right-left Yetter-Drinfeld modules, ${}^H{\mathcal{YD}}_H$, is a braided monoidal category. The monoidal structure and the braiding are given by $$\lambda(m1_{(1)}\ot n1_{(2)})=m_{[-1]}n_{[-1]}\ot m_{[0]}\ot n_{[0]};$$ $$\sigma_{M,N}(m1_{(1)}\ot n1_{(2)})=nm_{[-1]}\ot m_{[0]}.$$ As a braided monoidal category, ${}^H{\mathcal{YD}}_H$ is isomorphic to ${\mathcal{Z}}_l({\mathcal{M}}_H)$ and $({\mathcal{YD}}_H^H)^{\rm in}$.\
The antipode $S:\ H\to H^{\rm op,cop}$ is an isomorphism of weak Hopf algebras. Observe that the target map of $H^{\rm op,cop}$ is $\varepsilon_s$, and that its source map is $\varepsilon_t$. Thus $S$ induces an isomorphism between the monoidal categories ${}_H{\mathcal{M}}$ and ${}_{H^{\rm op,cop}}{\mathcal{M}}$. We also have a monoidal isomorphism $F:\ {}_{H^{\rm op,cop}}{\mathcal{M}}\to \ol{{\mathcal{M}}}_H$, given by $$F(M)=M,~~mh=h^{\rm op,cop}m.$$ indeed, in ${}_{H^{\rm op,cop}}{\mathcal{M}}$, $M\ot_t N$ is generated by elements of the form $1_{(2)}m\ot 1_{(1)}n$, and $F(M\ot_t N)$ is generated by elements of the form $m1_{(2)}\ot n1_{(1)}$. $F(N)\ot_sF(M)$ is generated by elements of the form $n1_{(1)}\ot m1_{(2)}$, and it follows that the switch map is an isomorphism $F(M\ot_t N)\to F(N)\ot_s F(M)$. We conclude from [Proposition \[pr:1.0.2\]]{} that we have isomorphisms of braided monoidal categories $${}_H^H{\mathcal{YD}}\cong {\mathcal{Z}}_l({}_H{\mathcal{M}})\cong {\mathcal{Z}}_l({}_{H^{\rm op,cop}}{\mathcal{M}})
\cong {\mathcal{Z}}_l(\ol{{\mathcal{M}}}_H)\cong \ol{{\mathcal{Z}}_r({\mathcal{M}}_H)}\cong \ol{{\mathcal{YD}}}_H^H.$$ This isomorphism can be described explicitely as follows: $$F:\ {}_H^H{\mathcal{YD}}\to \ol{{\mathcal{YD}}}_H^H,~~F(M)=M,$$ with $$m\cdot h=S^{-1}(h)m~~;~~\rho(m)=m_{[0]}\ot S(m_{[-1]}).$$ We summarize our results as follows:

[\[th:2.10\]]{} Let $H$ be a weak Hopf algebra with 