If $F:\pi_1(X)\rightarrow Ab$ is a system of local coefficiens, on the topological space $X$, then we can define the homology of $X$ with coefficients $F$ by taking the homology of the chain complex $C_p(X,F)=\bigoplus F(\sigma(e_1))$ where the sum is taken over all continuous $\sigma:\Delta^p \rightarrow X $, and the differentials are obvious (or more or less, care must be taken when computing $d_0\sigma$).
An alternative definition, in the case $X$ admits a universal cover $p:(\tilde X,\tilde x) \rightarrow (X,x)$, is to consider $S(\tilde X)$ as a left $\mathbb Z[\pi_1(X,x)]$ module ($\pi_1(X,X)$ is isomorphic to $Aut(p)$ , by taking $\gamma$ to the unique deck transformation that sends $\tilde x$ to $\tilde x \gamma$), and given $F$ a right $\mathbb Z[\pi_1(X,x)]$ module , we take the tensor product $F\otimes S(X)$ over $\mathbb Z[\pi_1(X,x)]$, and compute the homology of this complex.
If $F:\pi_1(X)\rightarrow Ab$, by restricting the functor to the automorphism group of $x$, which is ismorphic to the opposite of the fundamental group $\pi_1(X,x)$, we get a right $\mathbb Z[\pi_1(X,x)]$ module. Computing thus the homology in this two different ways, do we get the same? Are this definitions ok? Any reference where this is shown?
I could define a morphism from the first chain complex to the second, but I'm having trouble on defining one from the second chain complex to the first (hoping to eventually get an equivalence).
Silly me! If I'm not mistaken this is theorem 24.1 of Eilenberg "Homology of Spaces with Operators".