symmetric monoidal category
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 $s$):
$$\text{xymatrix@}+=2cmA\otimes I\text{xy@@ix@}$$