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.