category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
2-natural transformation?
A sylleptic monoidal 2-category is a monoidal 2-category whose E1-algebra structure is refined to an E3-algebra structure.
sylleptic monoidal 2-category, sylleptic 3-group