Great question! Yes, this is missing from the original paper and is certainly needed for the main result. It is true though. There's two different proofs currently in the literature, Deshpande and Mukhopadhyay Corollary 2.11 and Davydov and Nikshych proof of Theorem 8.5.
You can also prove it using an argument I learned from Theo Johnson-Freyd, namely given an object $X$ in $C_g$ you can realize $C_g$ as $A$-mod in $C_e$ with $X$ corresponding to $A$ itself, then $C_{g^{-1}}$ can be identified with mod-$A$ and the dual object will be $A$ again and the evaluation and coevaluation maps are given by multiplication and the unit of the algebra structure on $A$.