Trying to apply the Yoneda Lemma to Kleisli categories.

Yoneda.hs

{-# LANGUAGE RankNTypes #-}

module Yoneda (
    Yoneda (..),
    liftYoneda,
    lowerYoneda,
) where

import Control.Monad.Trans.Class (MonadTrans (..))

newtype Yoneda t m a = Yoneda {runYoneda :: forall r. (a -> m r) -> t m r}

liftYoneda :: (MonadTrans t, Monad m) => t m a -> Yoneda t m a
liftYoneda x = Yoneda $ \k -> x >>= (lift . k)

lowerYoneda :: (Monad m) => Yoneda t m a -> t m a
lowerYoneda y = runYoneda y return

instance Functor (Yoneda t m) where
    fmap f m = Yoneda $ \k -> runYoneda m (k . f)

instance (MonadTrans t, Monad m) => Applicative (Yoneda t m) where
    pure x = Yoneda $ \k -> lift (k x)

    f <*> x = Yoneda $ \k -> runYoneda f return >>= \g -> runYoneda x (k . g)

instance (MonadTrans t, Monad m) => Monad (Yoneda t m) where
    x >>= f = Yoneda $ \k -> runYoneda x return >>= \y -> runYoneda (f y) k