# 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. 1.

(unit coherence for $s$):

 $\xymatrix@+=2cm{A\otimes I{}{}{}{}{}{}{}{}{}{}\xy@@ix@{{\hbox{}}}}$