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Hello Noah, thanks for the reply! I actually wanted to work with the relative Tambara Tensor Product which had a similar description. And in your paper "The Balanced Tensor Product of Module Categories" with Douglas and Schommer-Pries, I saw a sentence saying that the relative Tambara Tensor product using Vect is equivalent to the Deligne Tensor Product.
@Adrien Thanks for the Help! I have actually be trying to learn your work "Integrating Quantum Groups over Surfaces" with Ben-Zvi and Jordan. I was wondering if given a braided monoidal 2-category, would one be able to get an $E_n$-algebra? And use the factorization homology framework? If the answer is in the positive, which $n$? What would be the analogue of $Pr_c$?
