I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories.

Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf algebra with the same multiplication, but the factors switched in the coproduct $\Delta$, and the antipode inverted. Both $R^{21}$ (switched factors) and $R^{-1}$ are $R$-matrix for this new coproduct.

Now, let $v$ be a ribbon element for $(A,R)$. Is $v^{-1}$ a ribbon element for $(A^{op},R^{21})$?

Checking a few examples suggests this is so, and I'm sure I could check it relatively easily. Is there anywhere I can reference?