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).