The Stacks project

Here's an argument for injectivity of $F'\to F$. Write $U_R$ for the presheaf 39.20.0.1. Since $U_R$ is separated (see #11670), the following Lemma says that injectivity of $F'\to F$ amounts to injectivity of $U_R\to F$. But the latter injectivity is clear.

Lemma. Let $\mathcal{C}$ be a site. Let $A$ be a separated presheaf of sets on $\mathcal{C}$ and write $A\to \tilde{A}$ for the sheafification of $A$. Let $B$ be a presheaf of sets on $\mathcal{C}$. A morphism of presheaves $\tilde{A}\to B$ is injective if and only if $A\to B$ is injective.

(I guess this Lemma could fit to either Section 7.10 or 7.11.)

Proof. ($\Rightarrow$). The morphism $A\to \tilde{A}$ is injective by Sites, Theorem 7.10.10, (2). Hence so is the composite $A\to\tilde{A}\to B$.

($\Leftarrow$). Let $s,t$ be sections of $\tilde{A}$ over $U\in\mathcal{C}$ that become equal when mapped into $B$. Let $\{U_i\to U\}$ be a cover of $U$ such that for each $i$, $s_{|U_i}$ and $t_{|U_i}$ come from sections of $A$ over $U_i$ (Sites, Lemma 7.10.16). Since $A\to B$ is injective, we deduce $s_{|U_i}=t_{|U_i}$. Thus $s=t$, for $\tilde{A}$ is a sheaf. $\square$