Let $X$ be a space which is paracompact, Hausdorff, and sufficiently nice that it has a universal covering space (and map) $p:\tilde{X}\to X$. Also, let $\pi:=\pi_1(X)$ and $A$ some $\mathbb{Z}[\pi]$-module, or if preferred, some abelian group on which $\pi$ acts. Under these conditions, there is a notion of cohomology with local coefficients which can be computed using cochain groups $C^n_\pi(X,A)=Hom_{\mathbb{Z}[\pi]}(C_n(\tilde X),A)$ where $C_n(\tilde{X})$ is the singular chain groups on $\tilde X$ and has an action of $\pi$ on it "inherited" from the action on $\tilde X$ itself by post-composition, that is, for $g\in \pi$, $\sigma \in C_n(\tilde X)$, $g\cdot \sigma$ is just $\sigma$ followed by the action of $g$. Let's call this cohomology $H^n_\pi(X,A)$.

On the other hand, there is another notion of cohomology with local coefficients whereby one defines a locally constant sheaf $\mathcal A$ on $X$ using the constant sheaf $A_\tilde{X}$ on $\tilde X$ and the action of $\pi$ on $A$, and takes the sheaf cohomology of that. It appears to be well known that (1) these two theories are essentially the same, and (2) for paracompact Hausdorff spaces, the Čech cohomology of a sheaf is the same as the sheaf cohomology of that same sheaf. That means that the Čech cohomology $\check{H}^n(X,\mathcal A)$ should be isomorphic to $H^n_\pi(X,A)$.

With all of this background out of the way, I can begin to describe my actual question. First of all, I cannot comprehend Sheaf cohomology, no matter how hard I try. Every reference I have found that describes it just seems impenetrable to me. On the other hand, I can more or less make sense of Čech cohomology, at least a lot better than "direct" sheaf cohomology. So: I would like to define some explicit map, $\check H^*(X ,\mathcal A)\to H^*_\pi (X, A)$ (or going the other way), which can exhibit this isomorphism more directly.

What appears natural for a map like this is to perhaps restrict to some convenient cofinal collection of covers of $X$ to get maps $C^*(\mathscr U,\mathcal A)\to C^*_\pi(X,A)$. A good candidate comes from the property of a universal cover, whereby every element $x\in X$ has a neighborhood $U_x$ so that $p^{-1}(U_x)$ composed of a disjoint union of open sets, each of which $p$ makes homeomorphic to $U_x$. It's not hard to show covers of the form $\mathscr U=\lbrace U_x : x\in X \rbrace$ with each $U_x$ a neighborhood like this is a cofinal collection, and then $p^{-1}(\mathscr U)$ is an open cover of $\tilde X$. There is a possibly useful property of Singular cohomology, too, which for any open cover $\mathscr V$ of $\tilde X$ makes the inclusion of the subcomplex $C_n(\mathscr V)=\langle \sigma: \text{im} (\sigma) \subset V\text{ for some }V \in \mathscr V \rangle$ a chain equivalence. That carries over to a cochain equivalence between $Hom_{\mathbb{Z}[\pi]}(C_n(\tilde X),A)$ and $Hom_{\mathbb{Z}[\pi]}(C_n(\mathscr V),A)$, and I'll call the latter group $C^n_\pi (\mathscr V , A)$.

My most promising candidate after several attempts, using these ideas, is to consider covers of $X$ as above, and for each one make a map $\psi_\mathscr{U}: \check{C}^n(\mathscr U , \mathcal A) \to C^n(p^{-1}(\mathscr U),A)$ by first applying a map $\iota_\mathscr{U}:\check{C}^n(\mathscr U , \mathcal A)\to \check{C}^n(p^{-1}(\mathscr U ), A_\tilde{X})$ defined by $(\iota_\mathscr{U} f)(p^{-1}(U_{x_0}),...,p^{-1}(U_{x_n}))=f(U_{x_0},...,U_{x_n})$, then picking out for each $\sigma \in C_n(p^{-1}(\mathscr U))$ the open set $U_\sigma:=U_{p(\sigma(v_0))}\cap ... \cap U_{p(\sigma(v_n))}$ and setting $(\psi_\mathscr{U} f)(\sigma):=[(\iota_\mathscr{U} f)(p^{-1}(U_\sigma))](\sigma(v_0))$. I say this is promising because evaluating at some point is needed in order to get the $\pi$-equivariance expected of an element of $C^n_\pi(p^{-1}(\mathscr U),A)$, and because it seems likely to give "injectivity" and "surjectivity" at least at the cohomology level after taking the direct limit defining Čech cohomology.

The only problem is that I'm not entirely sure this actually gives a cochain map. Running through the calculations, I've found that:

$(\partial\psi_{\mathscr{U}}f)(\sigma)-(\psi_{\mathscr{U}}\partial f)(\sigma)=[(\iota_{\mathscr{U}}f)(p^{-1}(U_{p(\sigma(v_{1}))}),...,p^{-1}(U_{p(\sigma(v_{n+1}))}))](\sigma(v_{1}))-[(\iota_{\mathscr{U}}f)(p^{-1}(U_{p(\sigma(v_{1}))}),...,p^{-1}(U_{p(\sigma(v_{n+1}))}))](\sigma(v_{0}))$

This is more or less because the '0 boundary' of the simplex sends $v_0$ to the same place the original simplex sends $v_1$ to, while the rest of the boundaries send $v_0$ to $\sigma(v_0)$. Of course, this needs to be $0$ to get a cochain map, but I'm not sure these two terms are the same (or if they are, how to show it).

So: Is this the wrong map again? If it is, what's the *right* map? Or, is there some really simple way to see that those two terms are the same and the difference really is $0$? Are there any references that have attempted something similar?