the definition that i proposed was overconstrained. it should be:
DEF. Let CC be a monoidal category. a *dagger* on CC is a functor P: CC^op ---> CC which is
* self-adjoint * equivalence * given together with the maps (*not* necessarily dinatural) ** e_X : X (x) PX ---> I ** h_X : I--->PX (x) X which make PX -| X.
if e_X and h_X happen to be dinatural, then P is just the star functor. in general, all we know about P is that it is a functor whose object part maps each object to some dual. the idea is that in the case of FHilb, we can take P to be the functor which will take a linear operator f:A-->B to the conjugate of the dual operator. to get the adjoint operator f^dag:B-->A, tranfer Pf:B*-->A* along the isomorphisms A~A* and B~B*, induced by the frobenius structures on A and B. sorry about the mistake, and thanks for pointing it out. -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]