z {\rangle}\\
& =\sum_{I, J\in\Lambda}{}_C {\langle}[\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}]\cdot E^X (W_I^{\rho}\cdot y) \, , \,
[S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}W_J^{\rho*}]\cdot E^X (W_J^{\rho}\cdot z) {\rangle}\\
& =\sum_{I, J\in\Lambda}[\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}]
{}_A {\langle}E^X (W_I^{\rho}\cdot y) \, , \, E^X (W_J^{\rho}\cdot z) {\rangle}[\psi_{(2)}\cdot_{\widehat{\rho}}W_J^{\rho}] \\
& =\sum_{I, J\in\Lambda}[\psi\cdot_{\widehat{\rho}}W_I^{\rho*} \, {}_A
{\langle}E^X (W_I^{\rho}\cdot y) \, , \, E^X (W_J^{\rho}\cdot z ) {\rangle}W_J^{\rho}] \\
& =\sum_{I, J\in\lambda}[\psi\cdot_{\widehat{\rho}} \, {}_C {\langle}W_J^{\rho*}\cdot E^X (W_I^{\rho}\cdot y) \, ,\, 
W_J^{\rho*}\cdot E^X (W_J^{\rho}\cdot z ) {\rangle}] \\
& =\psi\cdot_{\widehat{\rho}}{}_C {\langle}y, \, z {\rangle}.\end{aligned}$$

\[prop:leftcoaction\]With the above notations, the linear map from $Y$ to $Y\otimes H$ induced by the bilinear map $ ``\triangleright"$ from $H^0 \times Y$ to $Y$ defined by $$\psi\triangleright y =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot \phi(y)\cdot
(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau))$$ for any $y\in Y$, $\psi\in H^0$ is a left coaction of $H$ on $Y$ with respect to $(C, \widehat{\rho})$ satisfying that $$\psi\triangleright x = \epsilon^0 (\psi)x$$ for any $x\in X$, $\psi\in H^0$.

By Lemmas \[lem:fix\], \[lem:definition1\], \[lem:definition2\] and \[lem:definition3\], it suffices to show that $1^0 \triangleright y=y$ for any $y\in Y$. Indeed, by Remark \[remark:calculus\] and [@KT4:morita Lemma 5.4] $$1^0 \triangleright y=\sum_{I\in \Lambda}W_I^{\rho*}\cdot E^X (W_I^{\rho}\cdot y)=y$$ for any $y\in Y$.

We denote by $\mu$ the above left coaction induced by $``\triangleright"$.

A coaction of a finite dimensional $C^*$-Hopf algebra on 