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Yeah, with Joseph and an earlier version of Mark Wildon, I don't get why this question is not considered "too broad". Open any book on lattice theory and you find a bunch of open problems. Why wouldn't they all qualify?
@MorganRogers Maybe I should have said "notable" example or something similar, but it is interesting how rich categories of presheaves of monoids can be. I'd be interested to hear more about this thesis you propose. It's of course true for the base category $Set$; it's relevant here that all cocontinuous functors $Set \to Set$ are of the form $M \times -$ for some set $M$. It occurs to me that being simultaneously monadic and comonadic is very close to the same as having adjoints on both sides plus iso-reflecting, and that this condition is closed under composition.
@MorganRogers I thought you were asking about presheaves over a monoid. I'm not sure what it is you want. A monadic functor is comonadic iff it has a right adjoint, and so explicit uses of being simultaneously monadic and comonadic could be boiled down to this fact (which is sometimes useful, as for example in applications of the lex comonad theorem in topos theory). Such functors preserve and reflect all limits and colimits. Might you be interested in the special case of Frobenius monad and monoid?
@MorganRogers It's mentioned for instance in Mac Lane-Moerdijk's book on topos theory. But there are a number of interesting examples beyond the group case which is the best studied case. For example, the category of reflexive graphs is equivalent to $Set^M$ for a certain monoid $M$; see also ncatlab.org/nlab/show/graphic+category.
@PatrickMassot If you mean a reference for the filter functor: this has been known essentially forever, but here is one (which characterizes the algebras of the filter monad as continuous lattices): books.google.com/…
