For a quantum group (a quasitriangular Hopf algebra) $A$, it has a distinguished element in $A \otimes A$ called the universal R matrix in many texts (e.g. Kassel). This confuses me, because nowhere do I see any statement about uniqueness of $R$.
In my brief reading, it does not seem that $R$ is universal in the categorical sense, nor is it unique. This element just seems to generate R-matrices via representations over modules. Is this the sense in which "universal" is used?
To summarize my question, in what sense is $R$ universal, and in what sense is it unique? Is it really appropriate to call it the universal R matrix?