A monoidal category $\mathcal{C}$ with tensor product $\otimes$ is said to be symmetric if for every pair $A,B$ of objects in $\mathcal{C}$ , there is an isomorphism $$s_{AB}:A\otimes B\cong B\otimes A$$ that is natural in both $A$ and $B$ such that the following diagrams are commutative

1. (unit coherence for $s$ ):
   
2. (associativity coherence for $s$ ):
   
3. (inverse law):
   

Some examples and non-examples of symmetric monoidal categories:

• The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
• The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
• More generally, a category with finite products is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
• The category of bimodules over a ring $R$ is monoidal. However, this category is only symmetric monoidal if $R$ is commutative.

Remark. A symmetric monoidal category is a braided monoidal category such that the inverse law: $s_{BA}\circ s_{AB}=1_{A\otimes B}$ holds.