Two questions about local systems:
Given an algebraic group G over a perfect field $k$ and given a colection of vectorial spaces $(F_y)_{y\in G}$ why there is an unique local system $F$ such that the stalk of $F$ in $y\in G$ is $F_y$?
Given a local sistem $F$ and consider an morphism $f:G\rightarrow G$ (in my case is the frobenius morphism) why there is an unique isomorphism (up to scaling) $\phi$ such that: $\phi:(f)^*F \rightarrow F$