2 more general formulation of my question

# Terminology for a notion of "categories parameterized by agroupGanother(symmetricmonoidal)category"

Consider the category $Space(G)$ of $G$-spaces. For $H\subset G$, there is a forgetful functor from $Space(G)$ to $Space(H)$. Also, for an object $X$ in $Space(G)$ and another object $X'$ in $Space(G')$, $X\times X'$ is in $Space(G\times G')$.

Consider also the category of $Mod(G)$ of modules over $H_G(pt)$. For $H\subset G$, there is a forgetful functor from $Mod(G)$ to $Mod(H)$. Also for a module $M$ in $Mod(G)$ and another module $M'$ in $Mod(G')$, $M\otimes_{\mathbb{Z}} M'$ is in $Mod(G\times G')$.

My question is whether there is an established terminology for the categories $C(G)$ parameterized by a group $G$ satisfying the axioms abstracting those listed above:

• For $H\subset G$ there is a forgetful functor from $C(G)$ to $C(H)$
• For $G$, $G'$ there is a "multiplication functor" from $C(G)\times C(G')$ to $C(G\times G')$

satisfying various relations. The equivariant cohomology functor $H_G: Space(G) \to Mod(G)$ is compatible with these two operations. Is there a terminology for such a functor?

Update: Pondering more about it, I think it is better to formulate the concept as "categories $\mathcal{C}$ contravariantly parameterized by another (symmetric) monoidal category $\mathcal{X}$ ", i.e.

• for an object $X$ in $\mathcal{X}$, there is a category $\mathcal{C}(X)$
• for a morphism $f:X\to Y$ between objects in $\mathcal{X}$, there is a functor $f:\mathcal{C}(Y)\to\mathcal{C}(X)$
• for an object $X$ and $Y$ with its product $X\times Y$ in $\mathcal{X}$, there is a multiplication functor which defines, by $o_1\in \mathcal{C}(X)$ and $o_2\in\mathcal{C}(Y)$, an object $o_1\times o_2 \in \mathcal{C}(X\times Y)$.
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# Terminology for a notion of "categories parameterized by a group G"

Consider the category $Space(G)$ of $G$-spaces. For $H\subset G$, there is a forgetful functor from $Space(G)$ to $Space(H)$. Also, for an object $X$ in $Space(G)$ and another object $X'$ in $Space(G')$, $X\times X'$ is in $Space(G\times G')$.

Consider also the category of $Mod(G)$ of modules over $H_G(pt)$. For $H\subset G$, there is a forgetful functor from $Mod(G)$ to $Mod(H)$. Also for a module $M$ in $Mod(G)$ and another module $M'$ in $Mod(G')$, $M\otimes_{\mathbb{Z}} M'$ is in $Mod(G\times G')$.

My question is whether there is an established terminology for the categories $C(G)$ parameterized by a group $G$ satisfying the axioms abstracting those listed above:

• For $H\subset G$ there is a forgetful functor from $C(G)$ to $C(H)$
• For $G$, $G'$ there is a "multiplication functor" from $C(G)\times C(G')$ to $C(G\times G')$

satisfying various relations. The equivariant cohomology functor $H_G: Space(G) \to Mod(G)$ is compatible with these two operations. Is there a terminology for such a functor?