Jason Dusek jason.dusek at gmail.com
Wed Aug 6 16:56:08 EDT 2008

The problem as stated is to find the unit for the adjunction:

((- x A), (-)^A x A)

The latter functor takes an arrow f to (f . -) x id_A and does
the obvious thing for objects. The co-unit diagram is given
as:

B^A x A ---- eval_AB ----> B
^                       ^
|                       |
|                       |
curry(g) x id_A               g : C x A -> B
|                       |
|                       |
|                       |
C x A --------------------+

This diagram is somewhat puzzling, because it seems the second
functor has turned into (-)^A ! Continuing in with that, we
get a unit diagram like this:

C ---- magic ----> (C x A)^A
|                     |
|                     |
|                     |
curry(g)              (g . -)
|                     |
|                     |
|                     v
+------------------> B^A

So what is the magic? It is an arrow that takes a C to an
arrow that takes an A and makes the product C x A. I want to
write curry(C x A) but that is ridiculous looking. What's the
right notation for this thing?

--
_jsn