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I have a category with only two objects. These objects are just sets, each set has two elements in it. I take the morphisms as all endo-maps of the sets. There are no morphisms between the sets. I take an endo-functor,$F$ that maps one set to another and it is such that $FF = I$ where $I$ is the identity functor. I also map every function on set $S_1$ to its cousin on $S_2$. By that I mean, constant maps go to constant maps and the map $f$ such that $ff=I$ goes to his same kind of function on the other set. I hope this is not too hard to picture. Can anyone say if it will be easy to form a monad/comanad ie I will surely find the right natural transformations? Does anyone want to hazard a guess about the algebra/coalgebra for this monad?

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 If I correctly understand your description of $F$, there is no natural transformation, in either direction, between $F$ and $FF$. So there is no possible structure of monad or comonad on $F$. I'll vote to close, since the question isn't research level if I understood it correctly (and needs clarification if I didn't). – Andreas Blass May 4 2012 at 20:53
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 @Ben, since $FF=I$ and $F$ swaps the two objects and there are no maps between the two objects, there is no natural map $F\to FF=I$. In particular, $F$ is not a monad not a comodad in any way. – Mariano Suárez-Alvarez May 5 2012 at 7:29

closed as too localized by Andreas Blass, Tom Leinster, Dan Petersen, Martin Brandenburg, Bill Johnson May 5 2012 at 15:46

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