By the discussions in [@KT4:morita Section 2] and the proof of Rieffel [@Rieffel:rotation Proposition 2.1], there is an isomorphism $\Psi$ of $D$ onto $C$ defined by $$\Psi(d)={}_C {\langle}1_A \cdot d \, , \, 1_A {\rangle}$$ for any $d\in D$, where $A$ is a closed subspace of $Y$ and the unit element in $A$ is regarded as an element in $Y$. Then for any $a\in A$ $$\Psi(a)={}_A {\langle}1_A \cdot a \, , \, 1_A {\rangle}={}_A {\langle}a , \, 1_A {\rangle}=a .$$ Also, for any $d\in D$ $$\begin{aligned}
(E_1^{\rho, u}\circ\Psi)(d) & =E_1^{\rho, u}({}_C {\langle}1_A \cdot d \, , \, 1_A {\rangle})
={}_A {\langle}G^A (1_A \cdot d) \, , \, 1_A {\rangle}={}_A {\langle}1_A \cdot E_1^{\sigma, v}(d) \, , \, 1_A {\rangle}\\
& =E_1^{\sigma, v}(d) .\end{aligned}$$ Thus, we obtain the following lemma:

\[lem:equation\]With the above notations, we suppose that $A' \cap C={\mathbf C}1$. Then $\Psi$ is an isomorphism of $D$ onto $C$ satisfying that $$\Psi|_A ={{\rm{id}}}_A, \quad E_1^{\rho, u}\circ\Psi=E_1^{\sigma, v}$$

Let $\widehat{\rho}$ and $\widehat{\sigma}$ be the dual coactions of $H$ on $C$ and $D$ induced by the twisted coactions $(\rho, u)$ and $(\sigma, v)$, respectively and let $C_1 =C\rtimes_{\widehat{\rho}}H^0$ and $D_1 =D\rtimes_{\widehat{\sigma}}H^0$ the crossed products of $C$ and $D$ by the actions of $H^0$ 