system of $X$ and a morphism from $(X,L)$ to $(Y,M)$ is a pair $(f,\varphi )$ where $f:X\to Y$ is a continuous function and $\varphi $ is a natural transformation from $L$ to $M\circ f_*$. Here $f_*$ denotes the functor $f_*:\Pi (X)\to\Pi (Y)$ induced by $f$ sending $x$ to $f(x)$ and $\gamma $ to $f\circ \gamma $.

Let $G$ be a finite group and $\operatorname{Or}_{G}$ denote the [*orbit category*]{} of $G$ whose objects are orbits $G/H$ where $H$ is a subgroup of $G$. The morphisms of $\operatorname{Or}_{G}$ from $G/H$ to $G/K$ are $G$-maps between them where we consider the left cosets $G/H$ and $G/K$ as left $G$-sets. We denote the morphism from $G/H$ to $G/K$ which sends $H$ to $aK$ by $\hat{a}$.

\[defn:eqlocalcoefsystem\] Let ${\mathcal Top}$ denote the category of topological spaces and continuous maps. Let $X$ be a $G$-space. We define a contravariant functor $\Phi(X):\operatorname{Or}_{G}\to {\mathcal Top}$ which sends $G/H$ to $X^H$ and $\hat{a}$ to $a:X^K\to X^H$. A $G$-[*equivariant local coefficient system*]{} on $X$ is a contravariant functor $\underline{\, L} :\operatorname{Or}_{G}\to {\mathcal L}$ such that $F\circ  \underline{\, L} =\Phi(X)$ where $F:{\mathcal L}\to {\mathcal Top}$ is the forgetful functor which sends $(X,L)$ to $X$ and $(f,\varphi )$ to $f$. We will 