@Santana Afton I find it difficult to analogize, since in topology I can think with actual loops and universal cover but I cannot do it here in algebraic geometry.

@Denis Nardin I am studying with Stacks project and Tamas Szamuely's Galois groups and fundamental groups and both of them only have the definition with fiber functor.

I thought the fundamental group is the automorphism group $Aut(F_{\bar{x}})$ where $F_{\bar{x}}$ is the functor from finite etale morphisms over $X$ to sets by taking the geometric fiber of $\bar{X}$. Are these definitions equivalent?

Thank you, I also found theorem 11.4 in Milne's étale cohomology and it looks more accessible.

also since I am working with smooth projective $k$-scheme and smooth over a field implies normal, thus implies geometrically unibranch. We can use lemma 7.4.10, whose proof basically says that any geometric cover $Z$ of $X$ is a disjoint union of finite étale covers. Since the pullback of $F$ on $Z$ is trivial, the pullback of $F$ on each disjoint component is trivial too. Is that correct?

Can you please check if I understand this approach correctly? I have a local system $F$ on $X_{\acute{e}t}$ with values in a $k$-vector space $V$, also a local system in $X_{pro\acute{e}t}$. This correponds to a geometric cover $Y$, which is an étale $X$-scheme. The fiber of each point is $V$ as a $\pi_1^{proet}(X,x)$-set and the action of the group gives rise to a representation. Let $U$ be the kernel of the representation, then $\pi_1^{proet}(X,x)/U$ is also a $\pi_1^{proet}(X,x)$-set. This set corresponds to another geometric cover $Z$ and the pullback of $F$ on $Z$ is trivial.

Thank you very much, lemma 7.4.7. only works for local fields. Do we know anything about other like $\mathbb{C}$?

but proposition 6.15 is for topological spaces and the next one 6.16 is for connected scheme, but for locally constant sheaf of finite stalks, so if I work with k-vector spaces of characteristic 0, then it is infinite.

Can you specify where in Milne's étale cohomology? I couldn't find it. Does $k$ have to be a finite field for this to work or does it also work for $k$ of characteristic 0.