[nyssos]

> Not to mention you need monadic and nonmonadic versions of every higher order function (or so it feels like) - map / mapM

This is more a weakness of Haskell's standard library (which is despite its reputation not very friendly to category theory) than an inherent problem with monads. A more general `map` would look something like this

    class Category c => FunctorOf c f where
    map :: c a b -> c (f a) (f b)

    fmap :: FunctorOf (->) f => (a -> b) -> f a -> f b
    fmap = map

    type Kleisli m a b = a -> m b
    mapM :: (Monad m, FunctorOf (Kleisli m) f) => (a -> m b) -> f a -> m (f b)
    mapM = map

    type Op c a b = c b a
    contramap :: FunctorOf (Op (->)) f => (a -> b) -> f b -> f a
    contramap = map

    type Product c d (a,x) (b,y) = (c a b, d x y)
    bimap :: FunctorOf (Product (->) (->)) f => (a -> b, x -> y) -> f (a, x) -> f (b, y)
    bimap = map

    -- and so on 

but this would require somewhat better type-level programming support for the ergonomics to work out.