bijective antipode. Then we have the following isomorphisms of braided monoidal categories: $${}_H^H{\mathcal{YD}}\cong {{}_H{\mathcal{YD}}^H}^{\rm in}\cong \ol{{\mathcal{YD}}_H^H}
\cong \ol{{}^H{\mathcal{YD}}_H}^{\rm in}.$$

Yetter-Drinfeld modules are Doi-Hopf modules
============================================

[\[se:3\]]{} It was shown in [@cmz] that Yetter-Drinfeld modules (over a classical Hopf algebra) can be considered as Doi-Hopf modules, and, a fortiori, as entwined modules, and as comodules over a coring (see [@BW]). Weak Doi-Hopf modules were introduced by Böhm [@Bohm], and they are special cases of weak entwined modules (see [@CDG]), and these are in turn examples of comodules over a coring (see [@BW]). In this Section, we will show that Yetter-Drinfeld modules over weak Hopf algebras are special cases of weak Doi-Hopf modules. We will discuss the left-right case.

[\[pr:3.1\]]{} Let $H$ be a weak Hopf algebra with a bijective antipode. Then $H$ is a right $H\ot H^{\rm op}$-comodule algebra, with $H$-coaction $$\rho(h)=h_{(2)}\ot S^{-1}(h_{(1)})\ot h_{(3)}.$$

It is easy to verify that $H$ is a right $H\ot H^{\rm op}$-comodule and that $\rho(hk)=\rho(h)\rho(k)$. Recall that $H_t={{\rm Im}\,}(\varepsilon_t)={{\rm Im}\,}(\ol{\varepsilon}_t)$. The target map of $H^{\rm op}\ot H$ is $\ol{\varepsilon}_t\ot\varepsilon_t$. We now have $$\begin{aligned}
&&\hspace*{-2cm}
1_{[0]}\ot (\ol{\varepsilon}_t\ot\varepsilon_t)(1_{[1]})=
1_{(2)}1_{(1')}\ot \ol{\varepsilon}_t(S^{-1}(1_{(1)}))\ot 
\varepsilon_t(1_{(2')})\\
&=& 1_{(2)}1_{(1')}\ot S^{-1}(1_{(1)})\ot 1_{(2')}=\rho(1),\end{aligned}$$ where we used the fact that $S^{-1}(1_{(1)})\ot 1_{(2)}\in H_t\ot H_t$.

[\[pr:3.2\]]{} Let $H$ 