Suppose we have a monoidal category equipped with additional data that almost makes it a braided monoidal category except that only one of the hexagon identities
is satisfied.
Can we then prove the other hexagon identity?
If not, is there an explicit counterexample and can we prove the other identity under the additional
condition that the braiding is symmetric?



Does one of the hexagon identities imply the other one?
Consider the category of $A-$graded vector spaces (here $A$ is an abelian group) with 
obvious tensor product and trivial associator. Then each isomorphism $a\otimes b\to b\otimes a$ can be specified as a nonzero complex number $B(a,b)$. Now the first hexagon axiom says that the function $B(a,b)$ is linear in the first variable and the second axiom says that $B(a,b)$ is linear in the second variable. Clearly these conditions are independent. The symmetric braiding would correspond to skew-symmetric function $B(a,b)$
and linearity in one variable would imply bilinearity.


