Let $X$ be an object in a monoidal category $({\cal C}, \otimes)$, and $\gamma:X \otimes X \to X \otimes X$ a braiding (that is to say a morphism in ${\cal C}$ from $X \otimes X$ to itself that satisfies the braid relation $$ (\gamma \otimes \text{id}) \circ (\text{id} \otimes \gamma) \circ (\gamma \otimes \text{id}) = (\text{id} \otimes \gamma) \circ (\gamma \otimes \text{id}) \circ (\text{id} \otimes \gamma). $$

For any other object $Y$ in ${\cal C}$ that is isomorphic to $X$, via an isomorphism $f:X \to Y$, it is clear that $$ (f^{-1} \otimes f^{-1}) \circ \gamma \circ (f \otimes f) $$ defines a braiding for $Y$. Moreover, for any non-equivalent isomorphism $g:X \to Y$, we get a different braiding on $Y$.

Thus, a single braiding for an object in ${\cal C}$ can induce a number of different braidings on any other object in its isomorphism class (given the existence of a non-equivalent isomorphisms that is). Is there a canonical way to extend a braiding from an object to its isomorphism class?

Yang-Baxter operator$R: X \otimes X \to X \otimes X$. That's what Joyal and Street call it, anyway. – Todd Trimble♦ Sep 6 '12 at 0:14