Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the nerve of $\mathcal{U}$. I am going to describe two maps from the singular cohomology of $N$ to the singular cohomology of $X$ (with real coefficients). My question is: do these maps coincide? I would be very surprised if they don't, and in fact I think that I could carry through a complete proof of this fact. Nevertheless, the only ideas that came to my mind would probably lead to a rather painful and not very instructive proof, so I was wondering if there could be any result in the literature (or any clever idea) that could help me avoiding unnecessary calculations.
The first map is well-known: Any partition of unity $\{\rho_i\}_{i\in I}$ subordinate to $\mathcal{U}$ induces a map $f\colon X\to N$ defined by $f(x)=\sum_{i\in I} \rho_i(x)\cdot i$. Distinct partitions of unity induce homotopic maps, so $f$ induces a well-defined map $$ f^*\colon H^*(N)\to H^*(X) $$ which does not depend on $\{\rho_i\}_{i\in I}$.
The second map makes use of a double complex. First of all, for every topological space $Z$ let us denote by $C^n(Z)$ the module of real singular cochains on $Z$. When $U$ varies among the open subsets of $X$, the association $U\mapsto C^n(U)$ defines a presheaf on $X$, and we denote by $\mathcal{C}^n$ the associated sheaf, so that $\mathcal{C}^n(U)$ is the space of sections of $\mathcal{C}^n$ over $U$. Since $X$ is nice enough, the inclusion of complexes $C^*(X)\to \mathcal{C}^n(X)$ induces an isomorphism $$ \psi\colon H^*(X)\to H^*(\mathcal{C}^*(X))\ . $$ (Here and below we are using that the usual differential on singular cochains induces a differential on the sections of the associated sheaf).
Let us now denote by $C^{p,q}(X)$ the double Cech-singular complex defined as follows: $$C^{p,q}(X)=\prod_{i_0<\ldots<i_p} \mathcal{C}^q(U_{i_0}\cap\ldots\cap U_{i_p})$$ (so we are using the sheaf rather than the presheaf of singular cochains), where the horizontal differential $C^{p,q}(X)\to C^{p+1,q}(X)$ is the usual Cech differential, and $C^{p,q}(X)\to C^{p,q+1}(X)$ is the differential induced by the singular differential.
After adding the column $C^{-1,q}(X)=\mathcal{C}(X)$ with the obvious restriction map $C^{-1,q}(X)\to C^{0,q}(X)$, the lines of this double complex become exact. Therefore, there exists an isomorphism $$ \varphi\colon H^*(\mathcal{C}^*(X))\to H^*(C^{*,*}(X))\ , $$ where $H^*(C^{*,*}(X))$ denotes the cohomology of the double complex.
On the other hand, since the intersection of $(p+1)$ elements of $\mathcal{U}$ corresponds to a $p$-simplex in $N$, the inclusion of constant maps into $0$-cochains induces an inclusion $C^p_{simpl}(N)\to C^{p,0}(X)$, where we denote by $C^p_{simpl}$ the space of simplicial cochains. After composing with the canonical isomorphism $H^*(N)\cong H^*_{simpl}(N)$, this inclusion induces a map $$ j\colon H^*(N)\to H^*(C^{*,*}(X))\ . $$
I can now formulate my question: do the maps $$ f^*\colon H^*(N)\to H^*(X)\, ,\qquad \psi^{-1}\circ\varphi^{-1}\circ j\colon H^*(N)\to H^*(X) $$ coincide?
