For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\otimes_R M$ satisfying the cocycle condition. This can be thought of as saying something about "agreeing on intersections" since it's demanding that the two ways of tensoring $M$ up to an $S\otimes_R S$ module, i.e. either along the left unit or the right unit $S\to S\otimes_R S$ are equivalent. This is basically, I think, the dual of saying that it agrees on projections.

Another common way of phrasing descent data is to say that $M$ is an $S$-module which is also an $S\otimes_R S$-comodule, where $S\otimes_R S$ is a coring with structure map $\Delta:S\otimes_R S\to S\otimes_R S\otimes_R S\cong S\otimes_R S\otimes_S S \otimes_R S$ using the unit map of $S$.

I have written down some vague things about how these two are the same, but is there a functorial equivalence between the two categories of descent data? How does one obtain a comodule structure from the isomorphism $M\otimes_R S\cong S\otimes_R M$? And vice versa? And if you're feeling pedagogical, you might even mention how these two notions are equivalent to the notion of being a coalgebra for a certain comonad!

Thanks!