The condition linking the balancing to the braiding, where $\theta$ is the balance and $\beta$ is the braiding, is that $\theta_{x \otimes y}$ should be the composite of $\beta_{x,y}$, $\theta_y \otimes \theta_x$, and $\beta_{y,x}$.

A balanced monoidal category is a special case of a balanced pseudomonoid in a balanced monoidal bicategory?.

Properties

Every symmetric monoidal category is balanced in a canonical way; in fact, the identity natural transformation (on the identity functor of $B$) is a balance on $B$ if and only if $B$ is symmetric. Thus balanced monoidal categories fall between braided monoidal categories and symmetric monoidal categories. (They should not be confused with balanced categories, which are unrelated.)