A braided monoidal category is a monoidal category equipped with natural isomorphisms 
satisfying the two hexagon
identities. For a strict monoidal category these
identities read
 |
and
 |
In general, the associativity isomorphisms enter these formulas. The braiding is symmetric when 
(Etingof et al. 2015).
Vector spaces with 
give a symmetric example. Braided tensor
equivalences preserve the monoidal structure and these braidings, as required in
the module comparison for vertex operator algebras.
