I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider this scenario:
Let $A$ be a $K$-algebra ($K$ is a field), and let $H$ be a Hopf algebra over $K$. Assume $A$ is a right $H$-comodule algebra (with $A^{\text{co}H}=K$) (also called right $H$-extension)
At some point, it's stated that this implies that $A\otimes_K A$ is an $H$-comodule, using what it calls codiagonal coaction (it just says it's the formal dual of the usual diagonal action of $H$ on $A\otimes_K A$). After a bit of research, I found that the diagonal action is $$\begin{array}{rcl} (A\otimes_K A)\times M & \longrightarrow & A\otimes_K A\\ (a\otimes_K b,m)& \longmapsto & am\otimes_K bm \end{array}$$ Is it trivial then that $A\otimes_K A$ is $H$-comodule just because the dual preserves structure? Do I have to find some expression for the dual from that to check if the properties of the definition of comodule are verified? Any help or hint will be appreciated, thanks in advance.