TITLE: Nonbraided rigid monoidal category where left and right duals coincide
QUESTION [5 upvotes]: In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no braiding for the category?

REPLY [6 votes]: The simplest example is G-graded vector spaces where G is a non-abelian group.