Given a lisse $\mathbb{F}_{\ell^r}$-sheaf on a smooth curve $U$ defined ofer $\bar{\mathbb{F}_p}$, Katz says here, in p. 33, that $\mathcal{F}$ "becomes constant on some connected finite etale galois covering $E \rightarrow U$". Why is this true?
If $\mathcal{F}$ is just an etale sheaf of $\mathbb{F}_\ell$-modules, this is similar to Prop. 7.6 here, but I don't know how to conclude the statement for the lisse sheaf and I haven't found any reference explicit enough.
EDIT: $\mathcal{F}$ corresponds to a continuous homomorphism $\pi_1(U, \bar{x}) \rightarrow GL_r(\mathbb{F}_\ell)$, and the target is finite, so we get a surjection onto the image $\pi_1(U, \bar{x}) \rightarrow G$. Somehow this surjection should correspond to a Galois cover of $U$ that trivializes $\mathcal{F}$, but I'm not sure how to see this explicitely.