Let $G$ be an affine algebraic group over $\mathbb{C}$, and suppose that every irreducible rational representation of $G\times G$ is of the form $V\boxtimes W$ (exterior tensor product), where $V,W$ are irreducible rational representations of $G$.
Let $\mathcal{A}$ be the category of rational $G$-representations. I believe it is an abelian, symmetric, monoidal, rigid category (I'm not great with these adjectives so please do alert me if I'm mis-using them).
My question: is there a categorical way of describing the category of rational $G\times G$-representations in terms of $\mathcal{A}$? In other words, is there general construction we could apply to $\mathcal{A}$ which would produce the category of rational $G\times G$-representations?
For example, if $G$ is reductive then it seems the category of rational $G$-representations is equivalent to the additive envelope of $\mathcal{A}\otimes\mathcal{A}$.