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.**



Is a composite of (co)monadic adjunctions (co)monadic?
This is not true in general. Note that by passage to the opposite category, your question is equivalent to asking that a composite of monadic functors is monadic. A counterexample for that is the following:


The category $\mathbf{Cat}$ of small categories is monadic over the category $\mathbf{Grph}$ of (directed) graphs (with loops), and $\mathbf{Grph}$ is monadic over $\mathbf{Set}$. But there is no monadic functor from $\mathbf{Cat}$ to $\mathbf{Set}$. Details can be found on page 107 of "Toposes, Triples and Theories" by Barr and Wells.


