category with duals (list of them)
dualizable object (what they have)
The term twist or twisted is one of the hugely overloaded terms in math. Among the various meaning it may have is
A twist, or balance, in a braided monoidal category is a natural transformation from the identity functor on to itself satisfying a certain condition that links it to the braiding. A balanced monoidal category is a braided monoidal category equipped with such a balance.
The condition linking the balancing to the braiding, where is the balance and is the braiding, is that should be the composite of , , and .
Every symmetric monoidal category is balanced in a canonical way; in fact, the identity natural transformation (on the identity functor of ) is a balance on if and only if is symmetric. Thus balanced monoidal categories fall between braided monoidal categories and symmetric monoidal categories. (They should not be confused with balanced categories, which are unrelated.)