The Stacks project

Lemma 7.3.4. Let $\mathcal{C}$ be a category. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets on $\mathcal{C}$. There exists a unique subpresheaf $\mathcal{G}' \subset \mathcal{G}$ such that $\varphi $ factors as $\mathcal{F} \to \mathcal{G}' \to \mathcal{G}$ and such that the first map is surjective.

Proof. To prove existence, just set $\mathcal{G}'(U) = \varphi _ U \left(\mathcal{F}(U)\right)$ for every $U \in \mathop{\mathrm{Ob}}\nolimits (C)$ (and inherit the action on morphisms from $\mathcal{G}$), and prove that this defines a subpresheaf of $\mathcal{G}$ and that $\varphi $ factors as $\mathcal{F} \to \mathcal{G}' \to \mathcal{G}$ with the first map being surjective. Uniqueness is straightforward. $\square$

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