On defining *-autonomous categories

[Michael Barr <barr@math.mcgill.ca>]

I got a note from a student named Benjamin Jackson saying that someone had
told him that to define a *-autonomous category, you need only a symmetric
monoidal category and a contravariant involution * such that Hom(A @ B,C*)
= Hom(A,(B @ C)*).  Here "=" means natural equivalence and @ is the tensor
product.  Not only is this apparently true but that equivalence is needed
only for A = I, the tensor unit.  Of course, what is going on is that all
the coherence is built-in to the structure of a symmetric monoidal
category.

First define A --o B = (A @ B*)*.  Then the isomorphism above implies that
    Hom(A,B) = Hom(I @ A,B) = Hom(I,(A @ B*)*) = Hom(I,A --o B)
Next we see that
    (A @ B) --o C = (A @ B @ C*)* = A --o (B @ C*)* = A --o (B --o C)
and, applying Hom(I,-), that
       Hom(A @ B,C) = Hom(A,B --o C)
Also we have that
  A --o I* = (A @ I)* = A* and A --o B = (A @ B*)* = (B* @ A)* = B* --o A*
which gives the structure of a *-autonomous category with dualizing
object I*.

The trouble with this is that generally speaking it is the --o which is
obvious and the tensor is derived from it.  The coherences involving
internal hom alone are much less well-known, although they are included
in the original Eilenberg-Kelly paper in the La Jolla Proceedings.
Still I am surprised that I never noticed this before.