In Bott, Tu, one is asked to calculate the cohomology of the following sheaf:
Exercise 10.7 (Cohomology with twisted coefficients). Let $\mathscr{F}$ be the presheaf on $S^1$ which associates to every open set the group $\mathbb{Z}$. We define the restriction homomorphism on the good cover $\mathfrak{U} = \{U_0, U_1, U_2\}$ (Figure 10.1) by: $$\begin{align} \rho^0_{01} & = \rho^1_{01} = 1,\\ \rho^1_{12} & = \rho^2_{12} = 1,\\ \rho^2_{02} & = -1,\; \rho^0_{02} = 1, \end{align}$$ where $\rho^i_{ij}$ is the restriction from $U_i$ to $U_i \cap U_j$. Compute $H^*(\mathfrak{U}, \mathscr{F})$. (Cf. presheaf on an open cover, p. 142.)
Is this a local system, i.e. a locally constant sheaf?
I think it should be: $S^1$ is covered by $\mathfrak{U}$, and $\mathscr F(U_i)=\mathbb Z$. But how can I show $\mathscr F|_{U_i}=\underline {\mathbb Z}_{U_i}$?
Isn't it needed that all the restriction of $F|_{U_i}$ maps are the identity? This does not seem to be the case. Do I need to work with smaller subsets?
If it turns out that this is not a local system, how would the local system with monodromy $-1$ look instead?