I'm studying the book *Etale cohomology and the Weil conjecture* by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is the "sheafification" procedure in étale world): see pages 11-13.

Let $A$ be a commutative, unital, Noetherian ring and $\operatorname{Et}(A)$ the category
of étale extensions of $A$ (étale means that the extensions $A \to B$ are flat, unramified
$A$-algebras of finite type).

We follow the notation from the book
and call as an étale presheaf (of set, abelian groups,...; for sake of simplicity say sets) a covariant functor

$$ \mathcal{F}: \operatorname{Et}(A) \to (Set), (Ab),... $$

Now we want to associate to $\mathcal{F}$ another étale presheaf $\widetilde{\mathcal{F}}$ as follows:

Let $B \in \operatorname{Et}(A)$ and
$\mathcal{B}:= (B \to B_i, i \in I) $ an *étale cover of $B$*, that's a finite family
$(B \to B_i, i \in I) $ of étale maps , when the images of $\operatorname{Spec} B_i$
cover all elements of $\operatorname{Spec} B$.

(These form naturaly a category; a morphism $\mathcal{B} \to \mathcal{B}'$ between

$$ \mathcal{B}= (B \to B_i, i \in I), \mathcal{B}'= (B \to B_j', j \in J) $$

is given by a map $\sigma:J \to I$ inducing $B_{\sigma(j)} \to B_j'$ (datails can be worked out easily).

Now we associate to an etale cover $\mathcal{B}= (B \to B_i)$ the object $\mathcal{F}(\mathcal{B})$
defined as set of all $I$-tuples $(s_i)_{i \in I}$ with $s_i \in \mathcal{F}(B_i)$ such that
all $s_i$ fulfil the **compatibility condition**,

that is for every $i,j$ should hold

$$f^l_i(s_i)= f^r_j(s_j),$$

where the involved maps are
$f^l_i: \mathcal{F}(B_i) \to \mathcal{F}(B_i \otimes_B B_j)$
(respectively $f^r_j: \mathcal{F}(B_j) \to \mathcal{F}(B_i \otimes_B B_j)$) are
naturally induced by family of canonical maps $ l_i: B_i \to B_i \otimes_B B_j, b_i \mapsto b_i \otimes 1$,

(resp $ r_j: B_i \to B_i \otimes_B B_j, b_i \mapsto b_i \otimes 1$ )

We define the presheaf $\widetilde{\mathcal{F}}$ as direct limit over all coverings of $B$

$$ \tilde{\mathcal{F}}(B):= \varinjlim_{\mathcal{B}} \mathcal{F}(\mathcal{B})$$

(details like checking thats this is a functor etc. I omit here)

Now we come to my **Questions**. The book states some very important facts without
giving a reference. And I'm looking for proofs of several statements
there.

**Question 1a:** It is stated (p. 12) that for every $B \in \operatorname{Et}(A)$
and etale cover $\mathcal{B}= (B \to B_i)$ the induced map
$\widetilde{\mathcal{F}}(B) \to \widetilde{\mathcal{F}}(\mathcal{B})$
is injective (note that $\widetilde{\mathcal{F}}$ is in general still
a presheaf). How can it be proved?

**Question 1b:** Futhermore it is claimed that if for all $B$ and
$\mathcal{B}= (B \to B_i)$ the maps

$$ \mathcal{F}(B) \to \mathcal{F}(\mathcal{B}) $$

are *injective*, then $\widetilde{\mathcal{F}}$ is already an étale sheaf.
Why that's true? (Rmk.: combined with statement of Question 1a this
implies that the application of double tilde
$\widetilde{\widetilde{\mathcal{F}}}$ gives a sheaf)

**Question 2:** Why is the double tilde functor (= sheafification as
we saw above modulo the two questions) $\widetilde{\widetilde{\mathcal{F}}}$ is
exact? That is if we apply if to short exact sequence $0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0$ of étale presheaves we obtain a ses of sheaves?

(recall, that by definition a sequence $0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0$ of presheaves on site $\operatorname{Et}(A)$ is ses iff $0 \to \mathcal{A}(B) \to \mathcal{B}(B) \to \mathcal{C}(B) \to 0$ is ses for *every* $B \in \operatorname{Et}(A)$.)

Is $\widetilde{\mathcal{F}}$ exact as functor between etale presheaves?