I would like to thank all those that I replied so far. Quoting John Baez <baez@math.ucr.edu>:
Perhaps it would be good to pose a specific question. What would you like to know about pivotal 2-categories? Or are you mainly just looking for references?
To answer John, I would like to know what condition is required on left and right adjoints in a 2-category K to ensure that a string diagram representing a 2-morphism in K is invariant under topological deformation restricting to the identity on the boundary. I prefer not to use monoidal 2-categories, just ordinary 2-categories/bicategories. If I take a monoidal bicategory with duals and forget the monoidal structure will this be what I am after? 2-tangles clearly have the property I am looking for, but what if we adjoin some new 2-morphism A to 2-tangles. What condition would I need in order to ensure that any string diagram with the new morphism A was invariant under topological deformation?
I'd be curious to know what if any replies you received.
Aside from the replies that have been posted, I have also received a pointer to the paper "Introduction to linear bicategories" by Cockett, Koslowski, and Seely. The condition that *a=a* is studied in the context of linear bicategories and what are called cyclic adjoints. In particular, the discussion of cyclic mates seems to especially relevant. But I have not finished reading the paper and am still trying to understand what implications the `linear' in linear bicatgories will have on the ordinary bicategory case. Regards, Aaron