Norms of Generalized Mackey and Tambara Functors

Let $G$ be a finite group. A $G$-Tambara functor can be defined as a product-preserving functor $\mathcal{P}_G \to \mathsf{Set}$ (satisfying one additional condition), where $\mathcal{P}_G$ is a category that is constructed in a straightforward way from the category of finite $G$-sets. By replacing the category of finite $G$-sets with other categories, we obtain a more general notion of "Tambara functor". This more general notion subsumes the notion of motivic Tambara functors introduced by Bachmann. In this article, we extend a result of Hoyer about $G$-Tambara functors to this more general context.