I'll be rather general here, but I do not think the question is imprecise.

It is known that given a cotriple $\bot:\mathcal{C}\to\mathcal{C}$ we can construct a canonical simplicial object of $\mathcal{C}$. In general, are there qualities of a simplicial object that indicate that it is in fact the one associated to some cotriple? Or more concretely, given a simplicial object, are there properties that it can satisfy so that we can construct a comonad, or is all of the necessary information lost in the construction of such an object?

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Okay this edit is made months later, because I realized that my question is perhaps not exactly well posed. What I really want to know, and still haven't made time to actually investigate (because I don't think it'd be hard to play with, from a homological algebra point of view), is an answer to the question: Given a a functor $F:C\to S(C)$ when is it induced by a comonad (where by $S(C)$ I just mean simplicial objects in $C$)? We have one direction of a correspondence that takes comonads to simplicial functors on a category (perhaps with some restriction of the domain of the functor). Is there a way of going back?

and an object of C? – Tom Leinster Oct 12 '11 at 2:41intendedmonoidal structure. – Todd Trimble♦ Oct 12 '11 at 4:29