Characterization of étale locally constant sheaves over a normal scheme

Question

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...

Answer 1

(1) $ \mathcal F \to g_* g^* \mathcal F$ is an injection on the generic point.

Proof: In other words, the map $g^* \mathcal F \to g^* g_* g^* \mathcal F$ arising from the unit is an isomorphism. By a basic property of adjunctions, composing this with the map $g^* g_* g^* \mathcal F \to g^* \mathcal F$ coming from the counit gives the identity, so this map must be injective on stalks and the counit must be surjective on stalks so that their composition can be isomorphic on stalks.

(2) Let $\mathcal F$ be locally constant - in particular assume it is locally the constant sheaf associated to the set $\Lambda$. Then $g_* g^* \mathcal F$ is locally the constant sheaf associated to the set $\Lambda$.

Proof: Both $g_* $ and $g^*$ are local constructions so we may work étale-locally, and thus we may assume $\mathcal F$ is actually isomorphic to $\Lambda$. (Here we use that every étale open set in a normal variety is again normal.) Thus $g^* \mathcal F$ is $\Lambda$. So we must check that $g_* \Lambda$ is isomorphic to $\Lambda$. Recall that $\Lambda$ is the sheaf that sends an open set $U$ to the group $\Lambda$ raised to the set of connected components of $U$. There is a natural map $\Lambda \to g_* \Lambda$ arising from the natural map from the set of connnected components of $g^{-1}(U)$ to the set of connected components of $U$. This map is an isomorphism since $X$ is normal: $g^{-1}(U)$ is just the set of generic points of components of $U$ and each component has exactly one generic point.

Now if $\mathcal F$ is locally constant with finite stalks, this immediately implies $\mathcal F \to g_* g^* \mathcal F$ is an isomorphism, since at the generic point it's a map $\Lambda \to \Lambda$ which is injective and hence surjective, and then as you note since it's an map of locally constant sheaves which is an isomorphism at one point it's an isomorphism everywhere. If you want this for infinite sheaves it should be possible to do this by writing them as a limit of finite sheaves but I'm not sure if it's easier to make the limit play nice with the adjunction than to do this a different way.

This gives the "only if" direction (combined with what you already noted). For if, just note that if the Galois action on the stalk of $\mathcal F$ at $\eta$ factors through the étale fundamental group then $g^* \mathcal F$ is isomorphic to $g^* \mathcal L$ where $L$ is the locally constant sheaf corresponding to that representation of the étale fundamental group. So $\mathcal F \cong g_* g^* \mathcal F \cong g_* g^* \mathcal L$ is locally constant by the above.