Symmetric monoidal category

This article defines a notion of a category along with some additional structure.
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Definition

A symmetric monoidal category is a monoidal category $({\mathcal {C}},\otimes )$ equipped with a commutativity isomorphism: for every $A,B\in \operatorname {Ob} {\mathcal {C}}$ an isomorphism:

$\tau _{AB}:A\otimes B\to B\otimes A$

such that:

• $\tau$ is natural in both variables.
• For any objects $A,B$, $\tau _{BA}\circ \tau _{AB}$ is the identity map on $A\otimes B$.