category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A braided monoidal functor is a functor between braided monoidal categories that is a monoidal functor which respects the braiding on both sides.
Between symmetric monoidal categories a braided monoidal functor is the same as a symmetric monoidal functor.
braided monoidal functor
An exposition is in