I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:

Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.

Then the claim is that an étale sheaf $\mathcal{F}$ on $X$ with finite stalks is *locally constant*
(i.e., there exists an étale cover $(U_i \to X)_i$ such that each $\mathcal{F} \rvert_{U_i}$ is a constant sheaf, cf. §1, Prop. 1.1) if and only if the *unit map* $\mathcal{F} \to g_*g^*\mathcal{F}$ is an isomorphism and the Galois action by $\operatorname{Gal}(\overline{k(\eta)}/ k(\eta))$ on the stalk $\mathcal{F}_{\overline{\eta}}$ factors through the étale fundamental group
$\pi_1(X, \overline{\eta})$.

**Question:** How to see that this holds?

Few remarks:

$\pi_1(X, \overline{\eta})$ equals the Galois group of maximal unramified extension of $k(\eta)$; see p. 41.

it seems that the author suggests that one can use here the equivalence of categories between locally constant sheaves and finite $\pi_1(X, \overline{x})$-sets given by $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$, respectively $M \mapsto \mathcal{F}(U)= \operatorname{Hom}_{\pi_1} (\operatorname{Hom}_X(\overline{x}, U),M) $.

The implication that for $\mathcal{F}$ locally constant the $\operatorname{Gal}(\overline{k(\eta)}/ k(\eta))$-action on $\mathcal{F}_{\overline{\eta}}$ factors through $\pi_1(X, \overline{\eta})$ is standard, e.g., (tag/0DV4). Essentially, that's the equivalence above & independence of geometric(= base) point.

But I not understand how the part with the
unit map
$\mathcal{F} \to g_*g^*\mathcal{F}$ fits in the picture.
Especially, why local constancy of $\mathcal{F}$ implies
the unit map is an isomorphism and respectively why it's
reversely necessary alongside the assumption on $\mathcal{F}_{\overline{\eta}}$. In light of a result discussed here which appears to be maybe useful here, the main problem with the calculation of the stalks of $g_*g^*\mathcal{F}$ is how to control $g^*\mathcal{F}$. In general it's even not locally constant again. What do we know about it in this context which might help here to establish this characterization of locally constant sheaves?

Can this problem be reduced to case $\mathcal{F}$ via descent techniques, ie after passing to an etale trivialization $U_i \to X$ of $\mathcal{F}$ to deal with $\mathcal{F} \vert _{U_i} \to g_*g^*\mathcal{F} \vert _{U_i}$ simplifying(?) the situation? But then what is the term on the rhside?

**#UPDATE\progress:** About that the unit $\mathcal{F} \to g_*g^*\mathcal{F}$ is an iso assuming $\mathcal{F}$ is locally constant.

Keeping in mind the philosophy @Balazs suggest that the involved locally constant sheaves are "rigid" enough that the "whole information" is concentrated more less in a single stalk (...please correct me if I misunderstood your point) - the stalk & the action datum - and it suffice to reconstruct the whole sheaf, it seems reasonable to me that we need only to show two things:

(1) That the unit map is an isomorphism in the stalk at generic point; still not know how to manage it. Maybe that's exactly the point where the normality assumption is going to be exploited.

**UPDATE #2**(on point (1)) I think one part where normality of $X$ is exploited is to show that the stalk of $g_*g^*\mathcal{F}$ and $\mathcal{F}$ at geometric points coinside abstractly, compare again with this already linked above. But nevertheless, at this stage it's a priori not clear me why this should neccessarily imply that the unit map have to be an iso, or not?

(2) That $g_*g^*\mathcal{F}$ is locally constant; a pullback of a locally constant sheaf retain this property, and to same is true for pushforward assuming it is performed along an etale map, but $g$ is *not* etale as noticed in the comments below.

How else one can firsly deduce that $g_*g^*\mathcal{F}$ is locally constant (motivation: in order to simplify to check
that the unit map is iso it suffice to sheck it on a single stalk (+ action compatibility) - the generic point is the plausible candidate - assuming(!) we already know that both unvolved sheaves are locally constant...

Firstly,that it is an iso in the generic point, andsecondly, that $g_*g^*\mathcal{F} $ is locally constant. (the idea is so far I understand the philosophy correctly is that for locally constant essentially the whole story is told at only one stalk + action data; it seems natural to to work with stalk at generic point. $\endgroup$5more comments