> I have a question: are there interesting things I can learn about (co)monads by studying adjoint monotone functions composed this way or that way?

Galois connections are [adjoint functors](https://en.wikipedia.org/wiki/Adjoint_functors) for the category of preordered sets. You can show that whenever \\(F \dashv G\\) then \\(G \circ F\\) is a monad and \\(F \circ G\\) is a comonad.

We've been thinking about this in the [Categories for the Working Hacker](https://forum.azimuthproject.org/discussion/comment/16072/#Comment_16072) discussion group if you are interested.