My question comes from Nagami's "Dimension Theory". There's a bit of windup.
Let $X$ be a normal Hausdorff space, and $A\subseteq X$ a closed subset. For an open cover $\mathcal{U}$ of $X$ denote by $M_{\mathcal{U}}$ the nerve of $\mathcal{U}$, and by $N_{\mathcal{U}}$, the nerve of the corresponding cover of $A$. Each equipped with the weak topology.
If $\mathcal{U}=\{G_{\alpha}\}_{\alpha\in\Omega}$ is a locally finite cover of $X$ made up of cozero elements (each element coming equipped with a nonnegative function $f_{\alpha}:X\rightarrow\mathbb{R}$ for which $G_{\alpha}=f_{\alpha}^{-1}(0,\infty)$) there is a continuous map $f:X\rightarrow M_{\mathcal{U}}$ defined by
$$f(x)=\sum_{\alpha\in\Omega}\left(\frac{f_{\alpha}(x)}{\sum_{\beta\in\Omega}f_{\beta}(x)}\right)$$
For a given pair of spaces $(X,A)$ denote the $n$-dimensional Cech cohomology of the pair with coefficients in $G$ by $\check{H}^{n}(X,A:G)$.
The following Lemma is stated in Nagami's book.
Lemma: Let $(X,A)$ be a pair of paracompact spaces ($X$ normal and Hausdorff, $A$ closed in $X$) and let $\mathcal{U}$ be a locally finite cover of $X$ whose elements are all cozero sets. Let $f:(X,A)\rightarrow (M_{\mathcal{U}},N_{\mathcal{U}})$ be the continuous map defined as above. Then $f^{*}=p_{\mathcal{U}}:\check{H}^{n}(M_{\mathcal{U}},N_{\mathcal{U}}:G)\rightarrow\check{H}^{n}(X,A:G)$ where $p_{\mathcal{U}}$ is the projection map of $\check{H}^{n}(M_{\mathcal{U}},N_{\mathcal{U}}:G)$ (which is isomoprhic to its simplicial cohomology) into $\check{H}^{n}(X,A:G)$.
Nagami states that the proof of this result is simple and it is therefore omitted. As one can expect, I do not immediately see how to prove this Lemma. How is this Lemma proven? I am new to studying Cech cohomology, so I don't quite know how to think of the groups just yet. Any help would be appreciated.