TITLE: Is a composite of (co)monadic adjunctions (co)monadic?
QUESTION [5 upvotes]: I think this is probably elementary, but some searching (and asking on the chatroom) hasn't turned up a result. Could anyone point me to a reference for (or counterexample to) the following statement? 
Given categories $\mathcal{C}, \mathcal{D}, \mathcal{E}$ and  comonadic adjunctions $\mathcal{C} \rightleftarrows \mathcal{D}, \mathcal{D} \rightleftarrows \mathcal{E}$, the composite adjunction is comonadic.

REPLY [3 votes]: Here is a positive result: Let $\mathcal{C} \to \mathcal{D}$ and $\mathcal{D} \to \mathcal{E}$ be two monadic functors. Assume that the underlying functor of the first monad $\mathcal{D} \to \mathcal{D}$ preserves reflexive coequalizers (this happens quite often!). Then, the composition $\mathcal{C} \to \mathcal{D} \to \mathcal{E}$ is monadic, too. This is sketched here. (Does anybody know a canonical reference for this result?) There you can also find another counterexample by Jim Dolan.