A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups of $X$ with local coefficients $H_*(X,F)$ as done e.g. in chapter VI of Whitehead's book "Elements of homotopy theory". Therein, it is also shown that this construction yields functors $H_i$ from the category of spaces with local coefficients (a morphism between (X,F) and (Y,G) is a continuous map $f\colon X\rightarrow Y$ with a natural transformation $\eta\colon F\rightarrow f^*G$) to the category of abelian groups.
Suppose now that $X$ is path-connected and has a base point $x\in X$. Then the inclusion $\pi_1(X,x)\rightarrow \Pi(X)$ is an equivalence of categories, so we can pick an inverse functor $i\colon \Pi(X)\rightarrow \pi_1(X,x)$. If we have an module $F$ over $\pi_1(X,x)$ (which is the same as a functor $\pi_1(X,x)\rightarrow Ab$), we get a functor $\Pi(X)\rightarrow Ab$ by $F\circ i$.
Hence, we can define the homology groups of $X$ with local coefficients in the $\pi_1(X,x)$-module $F$ as $H_*(X,F):=H_*(X,F\circ i)$.
First question: Why is this definition independent of the chosen inverse $i$?
It seems to be a common thing to identify local coefficient systems over a path-connected space with modules over the fundamental group at a chosen point, so this has to be true.
Second question: Does this definition extend to a functor $H_i$ from the category of pointed path-connected spaces $(X,x)$ with modules over $\pi_1(X,x)$, where morphisms between $(X,x,F)$ and $(Y,y,G)$ are continuous maps $f\colon X\rightarrow Y$ together with an $\pi_1(f)$-equivariant map $F\rightarrow G$.
As the identification mentioned above is abundant in books and papers within algebraic topology, this should be true as well, but I encounter difficulties in proving this. The main problem seems to be that it might not be possible to choose the inverses $i\colon\Pi(X)\rightarrow \pi_1(X,x)$, such that they form a natural transformation.