I am reading [1] and trying to understand the identification of the cohomology with local coefficients $\bar{M}$ on a space $X$ and cohomology of $C_*(\tilde{X}) \otimes_G M$ with $M$ a $\mathbb{Z}[G]$-module. ($\tilde{X}$ is the universal covering of $X$ and $G = \pi_1(X,x_0)$).
I summarize the idea of the proof: $M$ is identified with $\bar{M}_{x_0}$, and a map $p: M \otimes_G C_*(\tilde{X}) \rightarrow C_*(X; \bar{M})$ is constructed in the following way.
For an element $g_0 \otimes \sigma \in M \otimes_G C_*(\tilde{X})$, $p(g_0 \otimes \sigma) = h_{p\circ \gamma}(g_0) \cdot (p\circ \sigma)$ where $g = h_{p\circ \gamma}: \bar{M}_{p(x_0)} \rightarrow \bar{M}_{p(y)}$ where $\gamma$ is a path connecting $x_0$ and $y = \sigma(e_0)$.
Then it is claimed that the it is not difficult to see map is an isomorphism. However, I am having issues convincing myself that the map is surjective. Namely, if $\sigma: \Delta^n \rightarrow X$ is a simplex with $n \geq 2$, then it admits a lifting to $\tilde{X}$. However, if $n = 1$, the lifting is just a path and I do not see how it can be the image of some $1$-simplex upstairs.
[1.] Whitehead, Elements of Homotopy Theory, pg. 278