Dear colleagues, I apologise for commenting a third time on this point. I think I can suggest a solution which could solve most of the problems mentioned in this line (not all of them), and can be used in various similar situations. We have a meaningful name for a concept, say involutive category (or category with involution). This term is even well established in category theory (see (*) below). Its main drawback is that the attribute "involutive" is not as flexible as we would like it to be, in order to develop a theory of such things: an involution-preserving functor cannot be called an "involutive functor", as Peter Selinger points out; and so on. On the other hand, typographical prefixes, like dagger or star, are already used in the same sense. They are flexible (you can say dagger category, dagger functor, and so on) and their authors do not want to give up the terminology of their previous works. Their big drawback - even forgetting about the fact that "category with involution" is an old, well established term - is that, being meaningless (or formalistic, as Walter Tholen prefers to say), nobody would search for them unless (s)he is already aware of this use. Thus, papers using such a terminology are often confined to some restricted domain; and this terminology will be reinvented again and again, with other acronyms or typographical signs. The solution I suggest: One can call such structures with a double name, a meaningful (possibly well established) name and a flexible one. In the present case this might be: - involutive categories (say), alias "prefix"-categories, - involution-preserving functors (say), alias "prefix"-functors, where "prefix" stays for the letter or acronym or typographical sign preferred by the author. In a title one would only use the meaningful terms, which can be easily retrieved in a search. In the body of a paper, one would use both terminologies in the main definitions and the short, adaptable prefix most of the time. --- (*) Searching in MathSciNet, under "Anywhere", I find: - category with involution: 41 items, starting with 1969 - involutive category: 5 items - dagger category: 0 items (including variants, like $\dagger$-category - star category: 1 item (all variants I can think of give the same item) (Of course, various dagger- or star- papers might be not yet on MathSciNet.) With best regards Marco Grandis