Some facts about sheafification functor on étale site

Question

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?

Answer 1

A good trick for answering questions of this type is that there are multiple foundational resources in étale cohomology. If a detail isn't explained in one you're reading, you can quickly check another one - the notation should hopefully be similar enough that you can transfer the proof over. In particular, the Stacks project tends to give a lot of details, and I found an answer to question 1 and a partial answer to question 2 there. (You should probably try to solve them as an exercise before doing this.)

The way I would say it.

Question 1a: Suppose we are given two elements of $\widetilde{\mathcal F}(B)$ whose image in $\widetilde{\mathcal F}(\mathcal B)$ is equal. We must show they are equal. By assumption, the two elements arise from covers $\mathcal A, \mathcal C$ of $\mathcal B$ and families of sections on these covers satisfying a compatibility condition. Their pullbacks to $\widetilde{\mathcal F}(\mathcal B)$ are given by the pullbacks of those families of sections to $\mathcal B \times_{B} \mathcal A$ and $\mathcal B\times_B \mathcal C$ respectively. If those are equal, then by the definition of direct limit, there is a common refinement where they are equal. But this common refinement is also a common refinement of $\mathcal A$ and $\mathcal C$ where our two families of sections are equal, so again by the definition of directly limit, the two starting elements of $\mathcal F(B)$ are equal.

Question 1b: Well, one must show that given a cover $U_1,\dots, U_n$ of $B$, the map from sections of $\widetilde{\mathcal F}$ on $B$ to compatible sections on $U_1,\dots, U_n$ is an isomorphism. So let's construct an inverse map. Given a compatible section of $\widetilde{\mathcal F}$ on a cover $U_1,\dots, U_n$, we obtain for each $i$ a cover $\mathcal U_i$ of $U_i$ and a compatible family of sections of $\mathcal F$ on that cover. We would like to make $\bigcup_i \mathcal U_i$ a cover of $B$ and this family of sections a compatible family. The only issue is that when we pass to an intersection $\mathcal U_i \times \mathcal U_j$, the pullbacks of the sections from $U_i$ and $U_j$ are only necessarily equal in $\widetilde{\mathcal F}$, not in $\mathcal F$. But given that $\mathcal F \to \widetilde{\mathcal F}$ is injective, we get the desired compatibility also in $\mathcal F$.

Question 2: No, $\tilde{F}$ is not exact as a functor between etale presheaves. To make a counterexample, take any exact sequence $0 \to \mathcal A \to \mathcal B \to \mathcal C \to 0$ of sheaves whose sequence of global sections $0 \to H^0( X, \mathcal A) \to H^0(X,\mathcal B)\to H^0(X,\mathcal C)$ is not right exact. Let $\mathcal C'$ be the quotient of $\mathcal B$ by $\mathcal A$ as presheaves, i.e. $\mathcal C'(U) = \mathcal B(U)/\mathcal A(U)$.

Then $0 \to \mathcal A \to \mathcal B \to \mathcal C' \to 0$ is a short exact sequence of presheaves. Its sheafification is simply $ \mathcal A \to \mathcal B \to \mathcal C$, which is a short exact sequence of sheaves, but not a short exact sequence of presheaves, since its global sections are not short exact.

Instead, the proof that sheafification is exact makes use of the fact that $\widetilde{\widetilde{\mathcal F}}$ is a left adjoint and therefore right exact, without proving right exactness for $\widetilde{\mathcal F}$.