Proof.
We have already seen in Lemma 101.3.3 that \Delta _ f is representable by algebraic spaces. Hence the statements (2) – (5) make sense, see Properties of Stacks, Section 100.3. Also Lemma 101.3.3 guarantees (2) holds. Let T \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} be a morphism and contemplate Diagram (101.3.3.1). By Algebraic Stacks, Lemma 94.9.2 the right vertical arrow is injective as a map of sheaves, i.e., a monomorphism of algebraic spaces. Hence also the morphism T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T is a monomorphism. Thus (3) holds. We already know that T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T is locally of finite type. Thus Morphisms of Spaces, Lemma 67.27.10 allows us to conclude that T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} \to T is locally quasi-finite and separated. This proves (4) and (5). Finally, Morphisms of Spaces, Proposition 67.50.2 implies that T \times _{\mathcal{X} \times _\mathcal {Y} \mathcal{X}} \mathcal{X} is a scheme which proves (1).
\square
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