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