In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a quasitriangular structure by definition.

Two pages further, Kassel defines a quantum invariant called the canonical $2$-tensor $t$, this definition involves the quasitriangular structure. Is there a categorical interpretation of this element? More precisely, given the category of (finite dimensional) representations of a quantum universal enveloping algebra, can I recover the element $t$ from this braided monoidal category? (The answer is yes if you are also given a fiber functor as you can then recover the Hopf algebra itself, so I'm not interested in that).