The Stacks project

Proof. Assume $\varphi$ is surjective. Pick $x \in X$. There exists an open neighbourhood $U$ of $x$ such that both $\mathcal{F}|_ U$ and $\mathcal{G}|_ U$ are isomorphic to $\mathcal{O}_ U^{\oplus r}$. Pick lifts of the free generators of $\mathcal{G}|_ U$ to obtain a map $\psi : \mathcal{G}|_ U \to \mathcal{F}|_ U$ such that $\varphi |_ U \circ \psi = \text{id}$. Hence we conclude that the map $\Gamma (U, \mathcal{F}) \to \Gamma (U, \mathcal{G})$ induced by $\varphi$ is surjective. Since both $\Gamma (U, \mathcal{F})$ and $\Gamma (U, \mathcal{G})$ are isomorphic to $\Gamma (U, \mathcal{O}_ U)^{\oplus r}$ as an $\Gamma (U, \mathcal{O}_ U)$-module we may apply Algebra, Lemma 10.16.4 to see that $\Gamma (U, \mathcal{F}) \to \Gamma (U, \mathcal{G})$ is injective. This finishes the proof. $\square$