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.
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.
As Daniel Schäppi pointed out, this is the same as asking whether the composite of two monadic functors is monadic. The answer, unfortunately, is no.
Consider locally presentable categories. By the classification theorem, every locally presentable category can be embedded as a reflective subcategory of some presheaf topos. It is not hard to see that a fully faithful functor with a left adjoint is automatically monadic. On the other hand, $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is monadic over $\mathbf{Set}^{\operatorname{ob} \mathcal{A}}$, as one might expect for a multi-sorted algebraic theory. Thus, for any locally presentable category $\mathcal{C}$, there exist a small category $\mathcal{A}$ and monadic functors $$\mathcal{C} \to [\mathcal{A}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Set}^{\operatorname{ob} \mathcal{A}}$$ but in general the composite $\mathcal{C} \to \mathbf{Set}^{\operatorname{ob} \mathcal{A}}$ is not monadic.
Indeed, it can be shown that any category monadic over $\mathbf{Set}^I$ (for some set $I$) must be an effective regular (= Barr-exact) category. But this is not true for a general locally presentable category, such as $\mathbf{Cat}$ or $\mathbf{Poset}$.
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.
