$$a_\psi\ot \Delta(c^\psi)=a_{\psi\Psi}\ot c_{(1)}^\Psi\ot c_{(2)}^\psi~~;~~
(ab)_\psi\ot c^\psi=a_\psi b_\Psi\ot c^{\Psi\psi};$$ $$1_\psi\ot c^\psi=\varepsilon(c_{(1)}^\psi)1_\psi\ot c_{(2)}~~;~~
a_\psi\varepsilon(c^\psi)=\varepsilon(c^\psi)a1_\psi.$$ Here we use the notation (with summation implicitely understood): $$\psi(a\ot c)=a_\psi\ot c^\psi.$$ An entwined module is a $k$-module $M$ with a left $A$-action and a right $C$-coaction such that $$\rho(am)=a_\psi m_{[0]}\ot m_{[1]}^\psi.$$ The category of entwined modules and left $A$-linear right $C$-colinear maps is denoted by ${}_A{\mathcal{M}}(\psi)^C$.\
Let $H$ be a weak bialgebra, and $A$ a right $H$-comodule, which is also an algebra with unit. $A$ is called a right $H$-comodule algebra if $\rho(a)\rho(b)=\rho(ab)$ and $1_{[0]}\ot \varepsilon_t(1_{[1]})=\rho(1)$.\
From [@Bohm], we recall the following definitions. Let $C$ be a left $H$-module which is also a coalgebra with counit. $C$ is called a left $H$-comodule algebra if $\Delta_C(hc)= \Delta_H(h)\Delta_C(c)$ and $${\label{eq:3.1.1}}
\varepsilon_C(hkc)=\varepsilon_H(hk_{(2)})\varepsilon_C(k_{(1)}c),$$ for all $c\in C$ and $h,k\in H$. Several equivalent definitions are given in [@CDG Sec. 4]. We then call $(H,A,C)$ a left-right weak Doi-Hopf datum. A weak Doi-Hopf module over $(H,A,C)$ is a $k$-module $M$ with a left $A$-action and a right $C$-coaction, satisfying the following compatibility relation, for all $m\in M$ and $a\in A$: $${\label{eq:3.1.2}}
\rho(am)=a_{[0]}m_{[0]}\ot a_{[1]}m_{[1]}.$$ The category of weak Doi-Hopf modules over $(H,A,C)$ and left $A$-linear right $C$-colinear maps is denoted by ${}_A{\mathcal{M}}(H)^C$.\
Let $(H,A,C)$ 