I've been thinking about generalising Functor, as I had something which was "Functor like" but didn't seem to fit into the existing definition. The example I'll present isn't the problem I was trying to solve, but it probably illustrates the solution better. The questions I have is: 1. Is this useful? 2. Has this been done? If the answers are Yes and No, I'll continue on into making this into a package. Anyway, here's what I've done: As we know, the type signature of fmap is the following: fmap :: Functor f => (a -> b) -> (f a -> f b) I've put brackets around the last two arguments to show that fmap can be also seen as a function which takes a function and returns a new function, in a different "space", for want of a better word. fmap should also follow this rule: fmap (f . g) == (fmap f . fmap g) So we can map an ordinary function into the "Maybe" space, but in a way that composing functions in the putting them in the space is the same as putting them in the space then composing them. I thought it would be nice if we could fmap to Kleisli Arrows, like so: fmap :: Monad m => (a -> b) -> Kleisli m a b Of course this is just "arr". But "arr" follows fmap's rules, so I thought it would be nice to make it a functor. So I was looking for more generalised versions of "Functor", and I found the 'categories' package, which had the following definition: class (Category https://hackage.haskell.org/package/base-4.7.0.1/docs/Control-Category.html#... r, Category https://hackage.haskell.org/package/base-4.7.0.1/docs/Control-Category.html#... t) => Functor f r t | f r -> t, f t -> r where fmap :: r a b -> t (f a) (f b) This solves half my problem, as now I can set r = (->) and t = Kleisli m and I've got the categories right. But it forces the "output" category to have a data constructor. So: fmap :: Monad m => (a -> b) -> Kleisli m a b Just won't fit. I'd need to make it: fmap :: Monad m => (a -> b) -> Kleisli m (Id a) (Id b) and then unwrap the results. I found this ugly though. So I've gone with another approached. I've linked the code here: http://ideone.com/TEg4MN and also included it inline at the bottom of this mail. Defining functors now becomes a bit wordy, but basically what I've defined is two functor instances. The first is just a copy of the existing functor instances to maintain the status quo behaviour. But secondly I've defined the Kleisli instance as discussed above. The "main" line, does two calls to fmap. The outermost (on the left) is just an ordinary fmap call on lists. The innermost, fmaps the function "triple" into a Kleisli Arrow, allowing it to be composed with the the Kleisli arrow already defined, 'evenOrNothing'. Type inference works this all out magically without signatures being required. Like I said, is approach new and useful? Improvements would be appreciated also, I'm only over the last few months really started focusing on learning Haskell properly (after leaving my previous job of 8 years), so I'm sure I'm still doing plenty of things not quite right. Thanks, Clinton --- {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE FlexibleContexts #-} module Main ( main ) where import qualified Control.Arrow import Prelude hiding (Functor, fmap, (.)) import Control.Category (Category, (.)) import qualified Data.Functor import qualified Control.Arrow import Control.Arrow (Kleisli(Kleisli), runKleisli) type family F (r :: (* -> * -> *)) (t :: (* -> * -> *)) x type family InputParam f x type family ResultParam f x type family InputCategory f :: (* -> * -> *) type family ResultCategory f :: (* -> * -> *) class Functor f where fmap :: ( Category r, Category t, f ~ F r t c, f ~ F r t d, r ~ InputCategory f, t ~ ResultCategory f, a ~ InputParam f c, b ~ InputParam f d, c ~ ResultParam f a, d ~ ResultParam f b ) => r a b -> t c d data OrdinaryFunctor :: (* -> *) -> * type instance F (->) (->) (f a) = OrdinaryFunctor f type instance InputParam (OrdinaryFunctor f) (f a) = a type instance ResultParam (OrdinaryFunctor f) a = f a type instance InputCategory (OrdinaryFunctor f) = (->) type instance ResultCategory (OrdinaryFunctor f) = (->) instance (Data.Functor.Functor f) => Functor (OrdinaryFunctor f) where fmap = Data.Functor.fmap data KleisliFunctor :: (* -> *) -> * type instance F (->) (Kleisli m) a = KleisliFunctor m type instance InputParam (KleisliFunctor f) a = a type instance ResultParam (KleisliFunctor m) a = a type instance InputCategory (KleisliFunctor m) = (->) type instance ResultCategory (KleisliFunctor m) = Kleisli m instance (Monad m) => Functor (KleisliFunctor m) where fmap = Control.Arrow.arr triple = (*3) evenOrNothing = Kleisli (\x -> if (even x) then Just x else Nothing) main = print $ fmap (runKleisli (evenOrNothing . fmap triple)) [3..6]