& =\sum_i (a\rtimes_{\sigma, v}1)x_i
\rtimes_{\widehat{\sigma}}\phi_i , \\
x(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 ) & =\sum_i (x_i \rtimes_{\widehat{\sigma}}\phi_i )
(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 ) \\
& =\sum_i x_i [\phi_{i (1)}\cdot_{\widehat{\sigma}}a\rtimes_{\sigma, v}1]\rtimes_{\widehat{\sigma}}\phi_{i(2)} \\
& =\sum_i x_i (a\rtimes_{\sigma, v}1)\rtimes_{\widehat{\sigma}}\phi_i .\end{aligned}$$ Since $(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 )x
=x(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 )$, $(a\rtimes_{\sigma, v}1)x_i =x_i (a\rtimes_{\sigma, v}1)$ for any $a\in A$ and $i$. Since $A' \cap D={\mathbf C}1$, $x_i \in {\mathbf C}1$ for any $i$. Therefore, $A' \cap D_1 \subset 1\rtimes_{\widehat{\sigma}}H^0$. The inverse inclusion is clear. Hence we obtain the conclusion.

By Lemma \[lem:Ciso\], $\widehat{\Psi}|_{A' \cap D_1}$ can be regarded as a $C^*$-algebra automorphism of $H^0$. We denote it by $\lambda^0$.

\[lem:Hopfiso\]With the above notations, $\lambda^0$ is a $C^*$-Hopf algebra automorphism of $H^0$.

It suffices to show that for any $\phi\in H^0$, $$S^0 (\lambda^0 (\phi))=\lambda^0 (S^0 (\phi)), \quad
\Delta^0 (\lambda^0 (\phi))=(\lambda^0 \otimes\lambda^0 )(\Delta^0 (\phi)), \quad
\epsilon^0 (\lambda^0 (\phi))=\epsilon^0 (\phi) .$$ We note that for any $h\in H$, $\phi\in H^0$, $$\phi(h)=N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}1)
(1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}e)
(1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}1)
(1\rtimes_{\widehat{\rho}}1^0\rtimes_{\widehat{\widehat{\rho}}}h))$$ by easy computations. Indeed, $$\begin{aligned}
& N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}1)
(1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}e)
(1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}1)
(1\rtimes_{\widehat{\rho}}1^0\rtimes_{\widehat{\widehat{\rho}}}h)) \\
& =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}e)
(1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}h)) \\
& =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi)[e_{(1)}\cdot_{\widehat{\widehat{\rho}}}
(1\rtimes_{\widehat{\rho}}\tau)]\rtimes_{\widehat{\widehat{\rho}}}e_{(2)}h) \\
& =N^2 E_2^{\rho, u}((1\rtimes_{\widehat{\rho}}\phi)[e_{(1)}\cdot_{\widehat{\widehat{\rho}}}
(1\rtimes_{\widehat{\rho}}\tau)])
\tau' (e_{(2)}h) \\
& =N^2 E_2^{\rho, u}((1\rtimes_{\widehat{\rho}}\phi)(1\rtimes_{\widehat{\rho}}\tau_{(1)})\tau_{(2)}(e_{(1)}))
\tau' 