TITLE: Monadic interpretation of coalgebras over operads
QUESTION [5 upvotes]: The structure of an algebra $A$ over a operad $O$ is encoded by an operad morphisms from $O$ to $\{Hom(A^{\otimes k},\, A)\}_{k}$. The same structure can be stored using the structure $M_OA\to A$ of an algebra over the monad $M_O$ which is defined on an object $A$ by $\bigoplus_j O(j)\otimes_{\Sigma_j} A^{\otimes j}$ and its structure maps $M_O\circ M_O\to M_O$ and $1\to M_O$ are induced from composition and unit respectively. 
The structure of an coalgebra $C$ over a operad $O$ is encoded by an operad morphisms from $O$ to $\{Hom(C,C^{\otimes k})\}_{k}$. 
Is there a monadic interpretation of coalgebras over an operad?

REPLY [2 votes]: What works is a comonadic interpretation of coalgebras over cooperads. Coalgebras over cooperads are like conilpotent coalgebras over operads.