[Haskell-cafe] Natural Transformations and fmap

[Daniel Fischer]

On Tuesday 24 January 2012, 04:39:03, Ryan Ingram wrote:
> At the end of that paste, I prove the three Haskell monad laws from the
> functor laws and "monoid"-ish versions of the monad laws, but my proofs
> all rely on a property of natural transformations that I'm not sure how
> to prove; given
> 
>     type m :-> n = (forall x. m x -> n x)
>     class Functor f where fmap :: forall a b. (a -> b) -> f a -> f b
>     -- Functor identity law: fmap id = id
>     -- Functor composition law fmap (f . g) = fmap f . fmap g
> 
> Given Functors m and n, natural transformation f :: m :-> n, and g :: a
> -> b, how can I prove (f . fmap_m g) = (fmap_n g . f)?

Unless I'm utterly confused, that's (part of) the definition of a natural 
transformation (for non-category-theorists).

> Is there some
> more fundamental law of natural transformations that I'm not aware of
> that I need to use?  Is it possible to write a natural transformation
> in Haskell that violates this law?
> 
>   -- ryan