The Stacks project

Proof. Choose an étale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$. Set $V = Y \times _ X U$. Denote $g : V \to Y$ and $j : V \to U$ the projections. Then $j : V \to U$ is a closed immersion and there is a unique point $v \in V$ mapping to $y \in |Y|$ and $u \in U$ (use Properties of Spaces, Lemma 66.4.3 and Spaces, Lemma 65.12.3). We have $j_*(\mathcal{G}|_ V) = (i_*\mathcal{G})|_ U$ as modules on the scheme $V$ and $j_*$ the “usual” pushforward of modules for the morphism of schemes $j$, see discussion surrounding Cohomology of Spaces, Equation (69.3.0.1). In this way we reduce to the case of schemes: if $i : Y \to X$ is a closed immersion of schemes, then

as modules over $\mathcal{O}_{X, x}$ where the module structure on the right hand side is given by the surjection $i_ y^\sharp : \mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$. Thus equality by Algebra, Lemma 10.52.5. $\square$