Let $X$ be a topological space, $R$ a ring, $R_X$ the locally constant sheaf defined via $$U \to \Gamma(U, R_X)=\{f:U \to R \vert f \text{ locally constant}\}$$
The $r$-th singular cohomology $C^r(X,R)$ is defined via
$$C^r(X,R) = \prod_{f:\Delta_r \to X}R = Hom(\bigoplus_{f:\Delta_r \to X}\mathbb{Z} \to R)$$
We are going to compare singular and sheaf cohomology.
In order to do this we define the presheaf $\bar{I_r}$
$$U\to C^r(U,R)$$
and get $I_r$ via sheafification of $\bar{I_r}$.
The coboundary maps
$$(\partial \phi)(f) = \sum(-1)^i\phi(..., f\vert _{\partial_i \Delta_r},...)$$
defines the complex
$$0 \to R_X \to I_0 \to I_1 \to ...$$
My question why is this complex exact if and only if $X$ is locally contractable?