of Corollary 4.8].

Let $A$ and $H$, $H^0$ be as before. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$. Let $C=A\rtimes_{\rho, u}H$ and $D=A\rtimes_{\sigma, v}H$. We also regard $A$ as an $A-A$-equivalence 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 $Y$ as a closed subspace. Furthermore, we suppose that $A'\cap C={\mathbf C}1$. Then by [@KT4:morita Lemma 10.3], $A' \cap D={\mathbf C}1$. Let $E_1^{\rho, u}$ and $E_1^{\sigma, v}$ be the canonical conditional expectations from $C$ and $D$ onto $A$ defined by $$E_1^{\rho, u}(a\rtimes_{\rho, u}h)=\tau(h)a , \quad
E_1^{\sigma, v}(a\rtimes_{\sigma, v}h)=\tau(h)a$$ for any $a\in A$, $h\in H$. By [@KT4:morita Theorem 2.7], there are a conditional expectation $F^A$ of Watatani index-finite type from $D$ onto $A$ and a conditional expectation $G^A$ from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $F^A$. Since $A' \cap D={\mathbf C}1$, by Watatani [@Watatani:index Proposition 4.1], $F^A =E_1^{\sigma, v}$. Hence $G^A$ is a conditional expectation from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $E_1^{\sigma, v}$. 