Edsko de Vries devriese at cs.tcd.ie
Tue Mar 4 13:30:05 EST 2008

```On Tue, Mar 04, 2008 at 11:58:38AM -0600, Derek Elkins wrote:
> On Tue, 2008-03-04 at 17:16 +0000, Edsko de Vries wrote:
> > Hi,
> >
> > Is there an intuition that can be used to explain adjunctions to
> > functional programmers, even if the match isn't necessary 100% perfect
> > (like natural transformations and polymorphic functions?).
>
> Well when you pretend Hask is Set, many things can be discussed fairly
> directly.
>
> F is left adjoint to U, F -| U, if Hom(FA,B) is naturally isomorphic to
> Hom(A,UB), natural in A and B.  A natural transformation over Set is
> just a polymorphic function. So we have an adjunction if we have two
> functions:
>
> phi :: (F a -> b) -> (a -> U b)
> and
> phiInv :: (a -> U b) -> (F a -> b)
>
> such that phi . phiInv = id and phiInv . phi = id and F and U are
> instances of Functor.
>
> The unit and counit respectively is then just phi id and phiInv id.

Sure, but that doesn't really explain what an adjunction *is*. For me,
it helps to think of a natural transformation as a polymorphic function:
it gives me an intuition about what a natural transformation is.