In the $G$-equivariant context for a finite group $G$, the role of abelian groups in non-equivariant algebra is now taken by Mackey functors. The category of Mackey functors is a closed symmetric monoidal category with symmetric monoidal product, the box product. Green functors are commutative monoids for this box product.

A Green functor is a Mackey functor$\underline{R}$ such that for all finite $G$-sets $T$, $\underline{R}(T)$ is commutative ring, such that all restriction maps are maps of commutative rings, and such that if $f: T\to T'$ is a map of finite $G$-sets, then we have the Frobenius reciprocity relation

$a \cdot T_f(b) = T_f(R_f(a) \cdot b)$

for all $a \in \underline{R}(T')$ and $b \in \underline{R}(T)$.