\[remark:calculus\]For any $\psi\in H^0$, $y\in Y$, $$\psi\triangleright y=\sum_{I\in \Lambda}[\psi\cdot_{\widehat{\rho}}W_I^{\rho*}]
\cdot E^X (W_I^{\rho}\cdot y) .$$ Indeed, since $1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}} \tau$ is the Jones projection for the conditional expectation $E_1^{\rho, u}$, for any $c\in C$ $(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)\cdot c=E_1^{\rho, u}(c)$. Also, $(1\rtimes_{\sigma, v}\rtimes_{\widehat{\sigma}}\tau)\cdot d =E_1^{\sigma, v}(d)$ for any $d\in D$. Hence by Lemma \[lem:pre\], for any $\psi\in H^0$, $y\in Y$, $$\begin{aligned}
&\psi\triangleright y \\
&=\sum_{I, J\in \Lambda}NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot W_I^{\rho*}
\otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*} )\otimes\widetilde{W_J^{\sigma*}}\cdot
(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\
& =\sum_{I, I_1 , J \in\Lambda}NE^Y ([\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]
E_1^{\rho, u}(W_{I_1}^{\rho}W_I^{\rho*})\otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*})
\otimes E_1^{\sigma, v}(W_J^{\sigma*})^{\widetilde{}}) \\
& =\sum_{I, I_1 , J\in \Lambda}NE^Y ([\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]
\otimes E_1^{\rho, u}(W_{I_1}^{\rho}W_I^{\rho*})\cdot E^X (W_I^{\rho}\cdot y\cdot W_{J}^{\sigma*})\cdot
E_1^{\sigma, v}(W_J^{\sigma})\otimes \widetilde{1}) \\
& =\sum_{I, I_1 , J\in\Lambda}[\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]\cdot E^X (E_1^{\rho, u}(W_{I_1}^{\rho}
W_I^{\rho*})W_I^{\rho}\cdot y\cdot W_J^{\sigma*}E_1^{\sigma, v}(W_J^{\sigma})) \\
& =\sum_{I_1 \in \Lambda}[\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]\cdot E^X (W_{I_1}^{\rho}\cdot y).\end{aligned}$$

\[lem:fix\]With the above notations, for any $x\in X$, $\psi\in H^0$, $\psi\triangleright x=\epsilon^0 (\psi)x$.

By Remark \[remark:calculus\], [@SP:saturated Theorem 2.2] and [@KT4:morita Lemma 6.9] $$\begin{aligned}
\psi\triangleright x & =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot\phi(x)\cdot
(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\
& =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi\tau)\cdot\phi(x)) \\
& =N\epsilon^0 (\psi)E^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)\cdot\phi(x))
=\epsilon^0 (\psi)\sum_{I\in\Lambda}[\tau\cdot_{\widehat{\rho}}W_I^{\rho*}]\cdot E^X (W_I^{\rho}\cdot x) \\
& =\epsilon^0 (\psi) ( \sum_{I\in\Lambda}E_1^{\rho, u}(W_I^{\rho*}E_1^{\rho, u}(W_I^{\rho})))\cdot x
=\epsilon^0 (\psi)x .\end{aligned}$$

\[lem:definition1\]With the above notations, for any $a\in A$, $h\in 