be a weak left-right Doi-Hopf datum, and consider the map $$\psi:\ A\ot C\to A\ot C,~~\psi(a\ot c)=a_{[0]}\ot a_{[1]}c.$$ Then $(A,C,\psi)$ is a weak left-right entwining structure, and we have an isomorphism of categories ${}_A{\mathcal{M}}(H)^C\cong {}_A{\mathcal{M}}(\psi)^C$.\
Let $(A,C,\psi)$ be a weak left-right entwining structure, and assume that $C$ is finitely generated projective as a $k$-module, with finite dual basis $\{(c_i,c_i^*)~|~i=1,\cdots,n\}$. Then we have a weak smash product structure $(A,C^*,R)$, with $R:\ C^*\ot A\to A\ot C^*$ given by $${\label{eq:1.41}}
R(c^*\ot a)=\hbox{$\sum_i$} \lan c^*,c_i^\psi\ran a_\psi\ot c_i^*.$$ We have an isomorphism of categories $${\label{eq:1.42}}
F:\ {}_A{\mathcal{M}}(\psi)^C\to {}_{\ol{A\#_R C^*}}{\mathcal{M}},$$ defined also follows: $F(M)=M$ as a $k$-module, with action $[a\# c^*]\cdot m=\lan c^*,m_{[1]}\ran am_{[0]}.$ Details can be found in [@CDG Theorem 3.4].

Yetter-Drinfeld modules over weak Hopf algebras
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[\[se:2\]]{} Let $H$ be a weak bialgebra. A left-left Yetter-Drinfeld module is a $k$-module with a left $H$-action and a left $H$-coaction such that the following conditions hold, for all $m\in M$ and $h\in H$: $$\begin{aligned}
&&\hspace*{-2cm}\lambda(m)= m_{[-1]}\ot m_{[0]}\in H\ot_t M;{\label{eq:2.1.1}}\\
&&\hspace*{-2cm} h_{(1)}m_{[-1]}\ot h_{(2)}m_{[0]}=(h_{(1)}m)_{[-1]}h_{(2)}\ot 
(h_{(1)}m)_{[0]}.{\label{eq:2.1.2}}\end{aligned}$$ We will now state some equivalent definitions. First we will rewrite the counit property for Yetter-Drinfeld modules.

[\[le:2.1\]]{} Let $H$ be a weak bialgebra, and $\lambda:\ M\to H\ot_t M$, $\rho(m)=m_{[-1]}\ot m_{[0]}$ a $k$-linear 