Hey Theo, sorry to bug you about this again, but I was thinking about it, and isn't the map $B\to B\otimes H$ always $A$-linear so long as we take the $A$-module structure on $B\otimes H$ given by the composition $A\to B\to B\otimes H$?

It seems strange to me that this question was closed as "too broad" while there is currently a question open asking for applications of sheaf theory to ring theory. I do not necessarily support the closing of that question, but point out that this question is significantly less broad than that other one.

Right, that last line can't be true. If that were the case then any colimit in a category with a zero object (e.g. pointed sets, abelian groups, spectra) would be trivial, which cannot be true.

Honestly I guess I can write down a whole bunch of diagrams and say the necessary things about them (good catch on the $A$-linearity by the way), but I kind of wonder why they don't seem to be built into the definitions. Also, I'm not necessarily working over a field (re: your statement about vector spaces).

Thanks Theo! I'd like to use as few commutativity conditions as possible, but it seems like there's a bit of give and take. The coring structure I had in mind was that $B\otimes_AB$ is a $B$-coring by $B\otimes_A A\otimes_AB\to B\otimes_A B\otimes_AB$, and $B\otimes H$ should be a coring by the diagonal map of $H$, i.e. $B\otimes H\to B\otimes H\otimes H\cong B\otimes H\otimes_B B\otimes H$.