I hope this is the right place to write this.

I was thinking about Fong and Spivak's first chapter, where they focus on [Galois connections](https://en.wikipedia.org/wiki/Galois_connection). Galois connections are a special case of [*adjunctions*](https://en.wikipedia.org/wiki/Adjoint_functors), where the functors are acting over two posets.

I thought it might be fun to do an exercise in Haskell based on Ed Kmett's [adjunction](https://hackage.haskell.org/package/computational-algebra-0.3.0.0/docs/Data-Functor-Adjunction.html) library.

In Haskell, we can write an adjunction between two _endofunctors_ as a type class. This type class has two natural transformations: leftAdjunct and rightAdjunct.


{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE TypeOperators #-}

module Adjunction where
import GHC.Generics

class (Functor f, Functor g) => Adjunction f g | f -> g, g -> f where
leftAdjunct :: (f a -> b) -> a -> g b
rightAdjunct :: (a -> g b) -> f a -> b


In category theory, this is written as \\(f \dashv g\\)

In this class, leftAdjunct and rightAdjunct are natural transformations and obey the following laws:


-- leftAdjunct . rightAdjunct === id
-- rightAdjunct . leftAdjunct === id


**Puzzle 1**: Give the type class instance for Adjunction ((,) e) ((->) e)

**Puzzle 2**: Define unit :: Adjunction f g => a -> g (f a) and counit :: Adjunction f g => f (g a) -> a such that:


-- leftAdjunct counit === id
-- rightAdjunct unit === id


**Puzzle 3**: Give a definition of leftAdjunct and rightAdjunct in terms of unit and counit (so now we can write {-# MINIMAL (unit, counit) | (leftAdjunct, rightAdjunct) #-} when we define the Adjunction type class).

**Puzzle 4**: Check out the [_functor composition type constructor_ (:.:)](https://hackage.haskell.org/package/base-4.11.0.0/docs/GHC-Generics.html#t::.:), and consider Comp1 . unit :: Adjunction f g => a -> (g :.: f) a. Write out instance Adjunction f g => Monad (g :.: f) where return = Comp1 . unit. What popular monad is this isomorphic to in the special case of Adjunction ((,) e) ((->) e)?