$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between $$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$

The proof in "Tensor Categories EGNO" uses the universal property of $\boxtimes$, but I would like to see an explicit functor that defines an equivalence.

In BaKi definition 1.1.15 there is a explicit description of $\mathcal{C}_1\boxtimes \mathcal{C}_2$ when $\mathcal{C}_1$, $\mathcal{C}_2$ are additive categories over $k$. Namely,

- $\operatorname{Ob}(\mathcal{C}_1\boxtimes \mathcal{C}_2)=$ finite sums of the form $\bigoplus X_i\boxtimes Y_i$, where $X_i\in \operatorname{Ob}(\mathcal{C}_1)$, $Y_i\in \operatorname{Ob}(\mathcal{C}_2)$.
- and $\operatorname{Hom}_{\mathcal{C}_1\boxtimes \mathcal{C}_2}(\bigoplus X_i\boxtimes Y_i,\bigoplus X_j'\boxtimes Y_j')= \bigoplus \operatorname{Hom}(X_i,X'_j)\otimes \operatorname{Hom}(Y_i,Y'_j)$.

Using this description I wanted to construct the equivalence.

I tried showing that \begin{align*} F:\Mod A\boxtimes \Mod B &\rightarrow \Mod{A\otimes_\mathbb{C} B}\\ X\boxtimes Y&\mapsto X\otimes_\mathbb{C}Y \end{align*} is full, faithul, and essentially surjective.

But I got stuck on trying to show that this functor is essentially surjective. There is what seems to be a counter example, shown on MathStack (Link).

Any help/suggestions would be greatly appreciated.

I currently only need the case when the modules are finite dimensional. But dont want to be restricted to finite dimensional algebras. Namely I am working with finite dimensional modules over $U(\mathfrak{g}_1\oplus \mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)$.

Edit (More context): I was really trying to use the Relative Tambara Tensor product introduced in (Tam01), and saw a sentence in (DSS18) (First Paragraph), that when using $\mathcal{C}=\operatorname{Vect}$ it agrees with Deligne's Tensor Product (though I could have misunderstood).

The Relative Tambara Tensor Product has a similar description of objects and morphisms, but also has much more relations.