Let $\mathcal{C}$ and $\mathcal{D}$ be two tensor categories (if necessary, assume they are fusion categories). Is the canonical braided monoidal functor $$\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(\mathcal{D})\rightarrow\mathcal{Z}(\mathcal{C}\boxtimes\mathcal{D})$$ an equivalence?


NB: The two monoidal categories $\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(\mathcal{D})$ and $\mathcal{Z}(\mathcal{C}\boxtimes\mathcal{D})$ have the same Frobenius-Perron dimension so it would be enough to show that the above functor is either injective or surjective in the sense of [EGNO](http://www-math.mit.edu/%7Eetingof/egnobookfinal.pdf).



Drinfeld center of a Deligne tensor product
By Cororllary 3.26 of arxiv:1009.2117, any braided tensor functor out of a non-degenerately braided fusion category is automatically fully faithful. Since $Z(\mathcal{C})\boxtimes Z(\mathcal{D})$ is non-degenerate when $\mathcal{C}, \mathcal{D}$ are fusion, the result follows immediately from your FP dimension observation.


