I had written:
My last comment is that, unlike what Jeff Egger claimed, "autonomous category" is not a special case of "*-autonomous category", because no symmetry is assumed in autonomous categories. Unless of course one first drops symmetry from the definition of *-autonomous categories, as Jeff has also suggested. As it stands, neither of "autonomous" and "*-autonomous" implies the other, which is perfectly fine in my opinion, since they are two different words.
I would like to clarify that Jeff himself did not say anything false, because in the context in which he said it, he had in fact assumed the non-symmetric definition of *-autonomous category (of [Barr 1995]). Sorry if it sounded like I was accusing him. My intention was only to point out that the statement "autonomous categories are a special case of *-autonomous categories" cannot be quoted out of context, because it is false under the original definition of *-autonomous category that includes symmetry (of [Barr 1979]). Since it had already been quoted out of context when I wrote the above, I just wanted to point out how the potential confusion. I think this is a very apt illustration of what happens if a term with an existing meaning is redefined to mean something else. Henceforth it is impossible for anybody to use the term (with either meaning) without first giving a definition. That's no problem in a math paper, where definitions are usually given or cited anyway, and therefore terminology is in principle arbitrary. But it does tend to hobble everyday discussion. -- Peter P.S.: since I have a demonstrated ability to put my foot in my mouth, I'd like to clarify that I am not accusing Mike Barr of anything either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous categories", and the inside of the paper clearly explains the distinction. It is only in subsequent use that any confusion arises. The usual solution, of putting either (non-symmetric) or (symmetric) in parentheses the first time the term is used, and omitting it for subsequent uses, is perfectly adequate. I am very happy with the statement "an autonomous category is a special case of a (non-symmetric) *-autonomous category". M. Barr (1979). "*-Autonomous Categories", Lectures Notes in Mathematics 752. Springer. M. Barr (1995). "Non-symmetric *-autonomous categories". Theoretical Computer Science 139:115–130. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]