$Y$, where $C_1 =C\rtimes_{\widehat{\rho}}H^0$ and $D_1 =D\rtimes_{\widehat{\sigma}}H^0$. As defined in [@KT4:morita], we define $Y_1$ as follows: We regard $C$ and $D$ as a $C_1 -A$-equivalence bimodule and a $D_1 -B$-equivalence bimodule in the usual way as in [@KT4:morita Section 4], respectively. Let $Y_1 =C\otimes _A X\otimes_B \widetilde{D}$. Let $E^Y$ be the conditional expectation from $Y_1$ onto $Y$ with respect to $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ defined by $$E^Y (c\otimes x\otimes\widetilde{d})=\frac{1}{N}c\cdot x\cdot d^*$$ for any $c\in C$, $d\in D$, $x\in X$, where $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ are the canonical conditional expectations from $C_1$ and $D_1$ onto $C$ and $D$ defined by $$E_2^{\rho, u}(c\rtimes_{\widehat{\rho}}\psi) =c\psi(e)  , \quad
E_2^{\sigma, v}(d\rtimes_{\widehat{\sigma}}\psi)=d\psi(e) .$$ for any $c\in C$, $d\in D$, $\psi\in H^0$, respectively. We regard $Y$ as a closed subspace of $Y_1$ by the injective linear map $\phi$ from $Y$ to $Y_1$ defined by $$\phi(y)=\sum_{I, J\in \Lambda}W_I^{\rho*}\otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*})
\otimes\widetilde{W_J^{\sigma*}}$$ for any $y\in Y$. In this section, we construct a left coaction of $H$ on $Y$ with respect to $(C, \widehat{\rho})$. First, we define the bilinear map $`` \triangleright "$ from $H^0 \times Y$ to $Y$ as follows: For any $\psi\in H^0$, $y\in Y$, $$\psi\triangleright y =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot
\phi(y)\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) .$$

