category with duals (list of them)
dualizable object (what they have)
Every diagram of constraint 2-cells in a free monoidal bicategory commutes; in other words, any two parallel composites of constraint 2-cells are equal. Moreover, any two parallel composites of constraint 1-cells are uniquely isomorphic.
Every monoidal bicategory is equivalent to a Gray-monoid.
The second version is a direct corollary of the coherence theorem for tricategories. The first can then be deduced from it (not entirely trivially).
Gordon, Power, Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558
Nick Gurski, Coherence in three-dimensional category theory