$D\subset D_1$ are also strongly Morita equivalent with respect to the $C_1 -D_1$-equivalence bimodule $Y_1$ and its closed subspace $Y$ by [@KT4:morita Corollary 6.3], we can see that the unital inclusions $D\subset D_1$ and $D\subset D\rtimes_{\beta}H^0 $ are strongly Morita equivalent with respect to the $D_1 -D\rtimes_{\beta}H^0$-equivalence bimodule $\widetilde{Y_1}\otimes_{C_1} (Y\rtimes_{\mu}H^0 )$ and its closed subspace $\widetilde{Y}\otimes_C Y\cong D$ by [@KT4:morita Proposition 2.2].

The exterior equivalence and the strong Morita equivalence {#sec:exterior}
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First, we shall present some basic properties on coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and their exterior equivalence and strong Morita equivalence.

Let $H$ and $K$ be finite dimensional $C^*$-Hopf algebras and let $H^0$ and $K^0$ be their dual $C^*$-Hopf algebras, respectively. Let $A$ and $B$ unital $C^*$-algebras and let $\pi$ be an isomorphism of $B$ onto $A$. Let $\lambda^0$ be a $C^*$-Hopf algebra isomorphism of $K^0$ onto $H^0$. Let $(\rho, u)$ be a coaction of $K^0$ on $B$ and let $\rho_{\pi, \lambda^0}$ be the homomorphism of $A$ to $A\otimes H^0$ defined by $$\rho_{\pi, \lambda^0}=(\pi\otimes \lambda^0 )\circ\rho\circ\pi^{-1} .$$ Let $u_{\pi, \lambda^0}$ be the unitary element in $A\otimes H^0 \otimes H^0$ defined by $$u_{\pi, \lambda^0}=(\pi\otimes\lambda^0 \otimes\lambda^0 )(u) .$$

\[lem:induced\]With the above notations, $(\rho_{\pi, \lambda^0} 