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