Tensor product of representations over a symmetric bimonoidal category

Definition
Suppose $$G$$ is a group and $$\mathcal{C}$$ is a symmetric bimonoidal category with operation $$\oplus$$ playing the role analogous to addition and operation $$\otimes$$ playing a role analogous to multiplication. Suppose $$A_1,A_2$$ are objects of $$\mathcal{C}$$ and $$\rho_1:G \to \operatorname{Aut}(A_1)$$ and $$\rho_2:G \to \operatorname{Aut}(A_2)$$ are group homomorphisms. We can thus think of $$(A_1,\rho_1)$$ and $$(A_2,\rho_2)$$ as representations of $$G$$ over $$\mathcal{C}$$. We define the tensor product of these representations, as a group homomorphism $$\rho_1 \otimes \rho_2:G \to A_1 \otimes A_2$$ where $$(\rho_1 \otimes \rho_2)(g) = \rho_1(g) \otimes \rho_2(g)$$ where the $$\otimes$$ on the right is the natural mapping $$\operatorname{Aut}(A_1) \times \operatorname{Aut}(A_2) \to \operatorname{Aut}(A_1 \times A_2)$$.