The Stacks project

Proof. We have already seen that $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ satisfies the sheaf property. Hence it remains to show the following: Given an fpqc covering $\{ T_ i \to T\} _{i \in I}$ of schemes over $B$ and an $\mathcal{O}_{X_ T}$-linear map $u : \mathcal{F}_ T \to \mathcal{G}_ T$ such that $u_{T_ i}$ is an isomorphism for all $i$, then $u$ is an isomorphism. Since $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$, see Topologies on Spaces, Lemma 73.9.3, this follows from Descent on Spaces, Proposition 74.4.1. $\square$