Dear Peter, thanks for your detailed response ! I agree with you an almost all points. First of all, I indeed just gave a method of constructing dagger categories out of some weaker structure, without solving the problem of giving a concept of dagger category stable under equivalence (this was not my intention !). Second, you are right that my analogy between star-pivotal and star-autonomous categories is a little awkward as stated in my previous message. What I wanted to say is that a pivotal category is an autonomous category in which one of the dualities is self-adjoint; and, "analogously", a star-pivotal category is a star-autonomous category in which homming into the dualizing object is self-adjoint. Concerning your point (b), I understand it only in part. "My" definition of a Hilbert space amounts to an isomorphism V->Hom(V,\bar C)=(V-bar)^* closed under self-adjunction. This is precisely a non-degenerate sesquilinear form on V with "hermitian symmetry", so one cannot do with much less. However, you are certainly right in saying that the existence of \bar C is something non-trivial. To finish, I wanted to tell you that I much appreciated reading your book on monoidal structures (Ross Street gave me a copy february 2009 when I was in Sydney). My initial motivation to learn about these came from joint work with Michael Batanin on the cyclic Deligne conjecture (on Hochschild cochains of a symmetric Frobenius algebra). From some formal considerations about operads, Michael and I came up with a categorical version of the Deligne conjecture, namely that the center of a pivotal (spherical) category is balanced (tortile), and we were convinced that, if true, such a statement was certainly known to specialists; I then gradually discovered that this was indeed the case and contained in work of Michael Mueger; however, I must admit that I still have no conceptual explanation of it. (I guess what I need is a deeper understanding of the trace(s) of a pivotal category). In any case, it was in thinking about pivotal categories as "categorification" of symmetric Frobenius algebras, that I began to play with closed involutive categories. All the best, Clemens.