u}1\rtimes_{\widehat{\rho}}\psi)(W_I^{\rho*}\rtimes_{\widehat{\rho}}1^0 )
(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) \\
& =\sum_{I\in\Lambda}([\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}\psi_{(2)})
(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) \\
& =\sum_{I\in\Lambda}([\psi\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}1^0 )
(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) .\end{aligned}$$ Therefore, we obtain the conclusion.

A left coaction of a finite dimensional $C^*$-Hopf algebra on an equivalence bimodule {#sec:left}
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Let $A$ and $B$ be unital $C^*$-algebras and let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. Let $A\rtimes_{\rho, u}H$ and $B\rtimes_{\sigma, v}H$ be the twisted crossed products of $A$ and $B$ by $(\rho, u)$ and $(\sigma, v)$, respectively. We denote them by $C$ and $D$, respectively. Then we obtain unital inclusions $A\subset C$ and $B\subset D$ of unital $C^*$-algebras. We suppose that $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$ in the sense of [@KT4:morita Definition 2.1]. Also, by [@KT4:morita Section 2], there is a conditional expectation $E^X$ from $Y$ onto $X$ with respect to $E_1^{\rho, u}$ and $E_1^{\sigma, v}$ satisfying Conditions (1)-(6) in [@KT4:morita Definition 2.2]. Furthermore, by [@KT4:morita Section 6], we can see that the unital inclusions $C\subset C_1$ and $D\subset D_1$ are strongly Morita equivalent with respect to the $C_1 -D_1$-equivalence bimodule $Y_1$ and its closed subspace 