(\psi_{(3)}^* \triangleright y) \, , \, 
S^0 (\psi_{(1)})\triangleright z {\rangle}\\
& ={}_C {\langle}S^0 (\psi_{(3)}^* )\triangleright T^* (\psi_{(4)}^* \triangleright y) \, , \, \psi_{(2)}S^0 (\psi_{(1)})
\triangleright z {\rangle}\\
& ={}_C {\langle}S^0 (\psi_{(1)}^* )\triangleright T^* (\psi_{(2)}^* \triangleright y) \, , \, z {\rangle}\\
& ={}_C {\langle}[S^0 (\psi^* )\rightharpoonup T^* ](y) \, ,\, z {\rangle}.\end{aligned}$$ Hence we can see that $[\psi\rightharpoonup T]^* =S^0 (\psi^* )\rightharpoonup T^* $. Therefore we obtain the conclusion.

By Lemma \[lem:action\], the map $``\rightharpoonup"$ is an action of $H^0$ on $D$. We denote by $\beta$ the coaction of $H$ on $D$ induced by the action $``\rightharpoonup"$ of $H^0$ on $D$. By the definition of the action $``\rightharpoonup"$, the left coaction $\mu$ of $H$ on $Y$ induced by the action $``\triangleright"$ is also a right coaction of $H$ on $Y$ with respect to $(D, \beta)$. Thus $\mu$ is a coaction of $H$ on $Y$ with respect to $(C, D, \widehat{\rho}, \beta)$. Hence we can see that $\widehat{\rho}$ is strongly Morita equivalent to $\beta$. By [@KT4:morita Section 4], the unital inclusions $C\subset C_1 (=C\rtimes_{\widehat{\rho}}H^0)$ and $D\subset D\rtimes_{\beta}H^0$ are strongly Morita equivalent with respect to the $C_1 -D\rtimes_{\beta}H^0$-equivalence bimodule $Y\rtimes_{\mu}H^0$ and its closed subspace $Y$. Since the unital inclusions $C\subset C_1$ and 