# [Haskell-cafe] Natural Transformations and fmap

Eugene Kirpichov ekirpichov at gmail.com
Tue Jan 24 06:43:53 CET 2012

```Have you tried generating a free theorem for :-> ? (I haven't as I'm writing from my phone)

24.01.2012, в 9:06, Ryan Ingram <ryani.spam at gmail.com> написал(а):

> On Mon, Jan 23, 2012 at 8:05 PM, Daniel Fischer <daniel.is.fischer at googlemail.com> wrote:
> 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).
>
> Alright, let's pretend I know nothing about natural transformations and just have the type declaration
>
> type m :-> n = (forall x. m x -> n x)
>
> And I have
> f :: M :-> N
> g :: A -> B
> instance Functor M -- with proofs of functor laws
> instance Functor N -- with proofs of functor laws
>
> How can I prove
>   fmap g. f :: M A -> N B
>   =
>   f . fmap g :: M A -> N B
>
> I assume I need to make some sort of appeal to the parametricity of M :-> N.
>
> > 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
>
>
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