Representation over a category

Definition
Suppose $$G$$ is a group and $$\mathcal{C}$$ is a category. A representation of $$G$$ over $$\mathcal{C}$$ is the following data $$(A,\rho)$$: an object $$A \in \operatorname{Ob} \mathcal{C}$$, and a homomorphism of groups $$\rho:G \to \operatorname{Aut}(A)$$.

We are generally interested in studying representations up to equivalence where two representations $$(\rho_1,A_1)$$ and $$(\rho_2,A_2)$$ of $$G$$ are termed equivalent if there exists an isomorphism $$\alpha:A_1 \to A_2$$ such that, for all $$g \in G$$:

$$\alpha \circ \rho_1(g) = \rho_2(g) \circ \alpha$$