Rigid Category
A monoidal category that is not symmetric, but otherwise obeys the duality axioms above, is known as a rigid category. A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is then a compact closed category.
Read more about this topic: Compact Closed Category
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