I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact:

The elements $s_{XY}$ of the $S$-matrix satisfy:

$\forall A,B\in \mathcal O(\mathcal C),\ s_{A^*B^*}=s_{AB}$

I tried to play around with the definitions:

$ s_{AB}=\text{Tr}(b_{BA}b_{AB}) := \text{Tr}^L(\psi_{A\otimes B}b_{BA}b_{AB}):=ev_{(A\otimes B)^*}\circ (\psi_{A\otimes B}b_{BA}b_{AB})\otimes id_{(A\otimes B)^*} \circ coev_{A\otimes B} $

$ s_{A^*B^*}=\text{Tr}(b_{B^*A^*}b_{A^*B^*}) := \text{Tr}^R(b_{B^*A^*}b_{A^*B^*}\psi^{-1}_{A^*\otimes B^*}):=ev_{(A^*\otimes B^*)^{\vee\vee}}\circ id_{(A^*\otimes B^*)^\vee}\otimes (b_{B^*A^*}b_{A^*B^*}\psi^{-1}_{A^*\otimes B^*}) \circ coev_{(A^*\otimes B^*)^\vee} $

But got nowhere.
Notation: $X^\vee$ is the right dual of $X$.


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