c(G/K)(\sigma )\, ) \text{\ \  in } L(G/H)(az_{\sigma })$$ for all $\sigma \in X^K$ and $a\in G$ with $a^{-1}Ha \leq K$. We can define a coboundary operator $\underline{\delta} :\Gamma ^{n-1}_G(X,\underline{L})\to \Gamma ^{n}_G(X,\underline{L})$ by the direct sum of the ordinary coboundary operators $$\underline{\, \delta }=\bigoplus _{H\leq G} \delta _H:\bigoplus _{H\leq G}\Gamma ^n(X^H ;L(G/H) )\to\bigoplus _{H\leq G}\Gamma ^{n+1}(X^H ; L(G/H) )$$ Since the ordinary coboundary operator is a natural transformation from $\Gamma^n$ to $\Gamma^{n+1}$ considered as functors from ${\mathcal L}$ to ${\mathcal Ab}$, we get $\underline{\, \delta }(\Gamma ^n_G(X;\underline{\, L} ))\subseteq \Gamma ^{n+1}_G(X;\underline{\, L} )$. The above definition easily generalizes to relative $G$-CW-complexes.

The $n$-th cohomology of a relative $G$-CW-complex $(X,A)$ with $G$-equivariant local coefficients $\underline{\, L}$ is defined as follows: $$H ^n_G(X,A;\underline{\, L} )=H^n\left(\Gamma ^*_G(X,A;\underline{\, L} ),\underline{\, \delta }\right).$$

We will be using the Bredon cohomology with a particular local coefficient system that comes from a $G$-fibration. We now introduce this coefficient system.

Let $p:E\to B$ be a $G$-fibration. Then $p^H:E^H\to B^H$ is a fibration for all $H\leq G$. Assume that for all $H\leq G$ and for all $b\in B$, the space $p^{-1}(b)^H$ is a path connected simple space. Associated to the fibration $p$, there is a $G$-equivariant local coefficient system 