Questions about braided monoidal categories
A braided monoidal category, or (“braided tensor category”), is a monoidal category $~\mathscr C~$ equipped with a natural isomorphism $$B_{x,y}:x⊗y→y⊗x$$ called the braiding, such that the following two kinds of diagrams commute for all objects involved (called the hexagon identities encoding the compatibility of the braiding with the associator for the tensor product):
$$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{B_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{B_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes B_{x,z}}{\to}& y \otimes (z \otimes x) }$$
and
$$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{B_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes B_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{B_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$$
where $~a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)~$ denotes the components of the associator of $~\mathscr{C}^\otimes~$.
