$A$ and its unitary element $\widehat{u}$ defined by $$h\cdot_{\rho, u}x=\widehat{\rho(x)}(h)=({{\rm{id}}}\otimes h)(\rho(x))$$ for any $x\in A$, $h\in H$. We call it the twisted action induced by $(\rho, u)$. Let $A\rtimes_{\rho, u}H$ be the twisted crossed product by the twisted action of $H$ on $A$ induced by $(\rho, u)$. Let $x\rtimes_{\rho, u}h$ be the element in $A\rtimes_{\rho, u}H$ induced by elements $x\in A$, $h\in H$. Let $\widehat{\rho}$ be the dual coaction of $H$ on $A\rtimes_{\rho, u}H$ defined by $$\widehat{\rho}(x\rtimes_{\rho, u}h)=(x\rtimes_{\rho, u}h_{(1)})\otimes h_{(2)}$$ for any $x\in A$, $h\in H$. Let $E_1^{\rho, u}$ be the canonical conditional expectation from $A\rtimes_{\rho, u}H$ onto $A$ defined by $$E_1^{\rho, u}(x\rtimes_{\rho, u}h)=\tau(h)x$$ for any $x\in A$, $h\in H$. Let $\Lambda$ be the set of all triplets $(i, j, k)$, where $i, j=1,2,\dots,d_k$ and $k=1,2,\dots,K$ with $\sum_{k=1}^K d_k^2 =N$. Let $W_I=\sqrt{d_k}\rtimes_{\rho, u}w_{ij}^k$ for any $I=(i, j, k)\in\Lambda$. By [@KT1:inclusion Proposition 3.18], $\{(W_I^{\rho *}, W_I^{\rho})\}_{I\in \Lambda}$ is a quasi-basis for $E_1^{\rho, u}$.

\[lem:pre\]With the above notations, for any $\psi\in H^0$, $$1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi
=\sum_{I\in\Lambda}([\psi\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}1^0 )
(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) .$$

Since $\{(W_I^{\rho*}, W_I^{\rho})\}_{I\in\Lambda}$ is a quasi-basis for $E_1^{\rho, u}$ by [@KT1:inclusion Proposition 3.18], $$\sum_{I\in\Lambda}(W_I^{\rho*}\rtimes_{\widehat{\rho}}1^0 )(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)
(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 )=1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}1^0 .$$ Hence for any $\psi\in H^0$, $$\begin{aligned}
1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi &
= \sum_{I\in \Lambda}(1\rtimes_{\rho, 