 equivalent to the twisted coaction $$\begin{aligned} (({{\rm{id}}}_{D_0}\otimes\lambda^0 )& \circ(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0 }) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-0} \, ,  \\ & ({{\rm{id}}}_{D_0}\otimes\lambda^0 \otimes\lambda^0 )((\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(v\otimes I_N ))) ,\end{aligned}$$ 