mal coaction and $\pi:A\to B$ a surjective homomorphism. Then $\pi$ is injective if and only if $\pi\vert_{A_e}$ is injective and there is a maximal coaction $\varepsilon$ of $G$ on $B$ such that $\pi$ is $\delta-\varepsilon$ equivariant.  The forward direction is immediate. Assume now that $\pi\vert_{A_e}$ is injective and there is a maximal coaction $\varepsilon$ of $G$ on $B$ such that $\pi$ is $\delta-\varepsilon$ equivariant. Then $\pi: (A, \delta)\to (B, \varepsilon)$ is a surjective morphism of coactions. Hence $\pi$ is an isomorphism by Proposition \[GIUTFB\], part .  The following is parallel to [Corollary ]{}:  \[cor: