Lemma 88.25.3. Let S be a scheme. Let f : X \to Y and g : Z \to Y be morphisms of algebraic spaces. Let T \subset |X| be closed. Assume that
X is locally Noetherian,
g is a monomorphism and locally of finite type,
f|_{X \setminus T} : X \setminus T \to Y factors through g, and
f_{/T} : X_{/T} \to Y factors through g,
then f factors through g.
Proof. Consider the fibre product E = X \times _ Y Z \to X. By assumption the open immersion X \setminus T \to X factors through E and any morphism \varphi : X' \to X with |\varphi |(|X'|) \subset T factors through E as well, see Formal Spaces, Section 87.14. By More on Morphisms of Spaces, Lemma 76.20.3 this implies that E \to X is étale at every point of E mapping to a point of T. Hence E \to X is an étale monomorphism, hence an open immersion (Morphisms of Spaces, Lemma 67.51.2). Then it follows that E = X since our assumptions imply that |X| = |E|. \square
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