 bimodule in the usual way. We suppose that the unital inclusions $A\subset C$ and $A\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $A$, that is, we assume that the $A-A$-equivalence bimodule $A$ is included in $Y$. Furthermore, we suppose that $A' \cap C={\mathbf C}0$. Then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $$(({{\rm{id}}}_