The Stacks project

Proof. Since $Z \amalg X' \to X$ is a surjective proper morphism we see that $\{ Z \amalg X' \to X\} $ is a ph covering (Topologies, Lemma 34.8.6). We have

Since $\mathcal{F}$ is a Zariski sheaf we see that $\mathcal{F}$ sends disjoint unions to products. Thus the sheaf condition for the covering $\{ Z \amalg X' \to X\} $ says that $\mathcal{F}(X) \to \mathcal{F}(Z) \times \mathcal{F}(X')$ is injective with image the set of pairs $(t, s')$ such that (a) $t|_ E = s'|_ E$ and (b) $s'$ is in the equalizer of the two maps $\mathcal{F}(X') \to \mathcal{F}(X' \times _ X X')$. Next, observe that the obvious morphism

is a surjective proper morphism as $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$. We conclude that $\mathcal{F}(X' \times _ X X') \to \mathcal{F}(E \times _ Z E) \times \mathcal{F}(X')$ is injective. It follows that (a) $\Rightarrow $ (b) which means that the lemma is true. $\square$