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, [Despande and Mukhopadhyay](https://arxiv.org/pdf/1909.10799.pdf) Corollary 2.11 and [Davydov and Nikshych](https://arxiv.org/pdf/2006.08022.pdf) 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 $A$.