Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl )

Question

I have question about a statement from Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15 and is seemingly a consequence of following two facts.

Let $A$ be a strict Henselian ring (i.e. Henselian + residue class field separably algebraically closed).
(This book works with definition 1.17, p 15, a noetherian local rings $A$ is Henselian if every local quasi-finite homomorphism from $A$ to a localization of a finitely generated $A$-algebra is finite).
Next, the authors make a remark and deduce a consequence:

*1.18 Remark.*Let A be a strictly Henselian ring. Then every local-etale homomorphism $A \to B$ is an isomorphism.
As a consequence it can be deduced from this that For every faithfully flat etale homomorphism $\varphi: A \to B$ of a strictly Henselian ring $A$ into a finitely generated $A$-algebra $B$ there exist a right inverse; that is a map $\psi:B \to A$ with $\psi \circ \varphi= id_A$ ( here an expanation why this consequence is true)

Now comes a claim I not understand. Next is said that for any sheaf of abelian groups $\mathcal{F}$ the functor

$$ \operatorname{Et}(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $$

is exact. Why that's true? (here $\operatorname{Et}(B)$ is the category of etale extensions of $B$ understood as full subcategory of the category of $A$-algebras)

Question 1:
My first confusion is that originally in the book the top line defining the functor reads as

$$ \operatorname{Et}(B) \to \operatorname{(Ab)}, \ B \mapsto \mathcal{F}(B) $$

This not make any sense, so obviously the author meant something else. What?
I guess (but not know) that the authors maybe intended to write $A$ instead of $B$, i.e.maybe to work with $\operatorname{Et}(A)$ etale extensions of a strict Henselian base; but I don't know, that's only a guess of mine, which "looks" for me more "natural". Does anybody see what the authors had there in mind?

Question 2: Why the statement about the exactness of the functor above is true?
We have to show that for every exact sequence

$$ 0 \to R_1 \to R_2 \to R_3 \to 0 $$

of etale $B$-algebras die Sequenz of abelain groups

$$ 0 \to \mathcal{F}(R_1) \to \mathcal{F}(R_2) \to \mathcal{F}(R_3) \to 0 $$

is exact too. Why? My first guess was that we can tensoring the $R_i$ by $- \otimes_B A$ via the right inverse $\psi$ from above and exploit then a splitting property of $A \to B$ for above remark, in order to show somehow that maybe $R_1 \to R_2$ has left inverse (or $R_2 \to R_3$ right inverse), but I not know how or if it really holds.

Answer 1

I think they meant to say that the functor $\mathcal F \mapsto \mathcal F(A)$ from sheaves on $A$ to abelian groups is exact.

The proof is:

If $\mathcal F_2\to \mathcal F_3 \to 0$ is an exact sequence, we need to show $\mathcal F_2(A) \to \mathcal F_3(A)$ is surjective.

Let $s \in \mathcal F_3(A)$ be a section. By definition of a surjective map of sheaves, there exists a covering $B$ of $A$ such that the pullback of $s$ to $B$ lies in the image of $\mathcal F_2(B) \to \mathcal F_3(B)$.

By the previous line, we can express $A$ as an open subset of $B$, so the pullback of $s$ to $\mathcal F_3(A)$, which is $s$ again, lies in the image of $\mathcal F_2(A) \to \mathcal F_3(A)$, as desired.