This question came up while reading about the equivalence between smooth $\mathbb{Q}_{\ell}$-sheaves and finite dimensional representations of $\pi_{1}(X,s)$. For a brief description of the situation, see Milne's LEC 19 page 124 in the section on "Sheaves of $\mathbb{Z}_{\ell}$-modules".
I am aware that taking stalks at $s$ is an equivalence of categories between locally constant sheaves of abelian groups with finite stalks and finite (continuous) $\pi_{1}(X,s)$-modules. Can anyone explain how the process of taking inverse systems of such modules, i.e. a smooth $\mathbb{Z}_{\ell}$-sheaf, results in a finitely generated $\mathbb{Z}_{\ell}$-module with a continuous $\pi_{1}(X,s)$-action? I find references to this fact everywhere, but no direct citations, and no proofs.
I'm most confused about the "finitely generated" part. If you look at footnote #35 on the same page in Milne, this is why.
For anyone who wants to know, the details can be found in Lei Fu's book Etale Cohomology, chapter 10, especially 10.1.4