[flebron]

I've always found the definition of monoid objects in a category of endofunctors to be tougher to grasp than the Kleisli-category definition of a monad, at least if one is being formal about "monoid object", and checking that the necessary diagrams commute.

Given a functor m of Hask, we define a squiggly arrow ~>, where a ~> b is a -> m b. If these ~> are the arrows in some category (which we call the Kleisli category for m), then we call m a monad. Here id :: a ~> a in that category is what we call return :: a -> m a, and the composition (.) :: (b ~> c) -> (a ~> b) -> (a ~> c) in that category is used to create bind :: m a -> (a -> m b) -> m b, where bind x f = (.) f (const x) (return ()), where (.) is the squiggly arrow (.) we just mentioned. Bind is what's spelled >>= in Haskell, and is what's behind the "x <- f" do-notation.