Proof.
Let U \to X be a surjective étale morphism from a scheme towards X. Then U \to X is representable, surjective, flat and locally of finite presentation by Morphisms of Spaces, Lemmas 67.39.7 and 67.39.8. By Lemma 80.4.3 the composition U \to F is representable by algebraic spaces, surjective, flat and locally of finite presentation also. Thus we see that R = U \times _ F U is an algebraic space, see Lemma 80.3.7. The morphism of algebraic spaces R \to U \times _ S U is a monomorphism, hence separated (as the diagonal of a monomorphism is an isomorphism, see Morphisms of Spaces, Lemma 67.10.2). Since U \to F is locally of finite presentation, both morphisms R \to U are locally of finite presentation, see Lemma 80.4.2. Hence R \to U \times _ S U is locally of finite type (use Morphisms of Spaces, Lemmas 67.28.5 and 67.23.6). Altogether this means that R \to U \times _ S U is a monomorphism which is locally of finite type, hence a separated and locally quasi-finite morphism, see Morphisms of Spaces, Lemma 67.27.10.
Now we are ready to prove that \Delta _ F is representable. Let T be a scheme, and let (a, b) : T \to F \times F be a morphism. Set
T' = (U \times _ S U) \times _{F \times F} T.
Note that U \times _ S U \to F \times F is representable by algebraic spaces, surjective, flat and locally of finite presentation by Lemma 80.4.4. Hence T' is an algebraic space, and the projection morphism T' \to T is surjective, flat, and locally of finite presentation. Consider Z = T \times _{F \times F} F (this is a sheaf) and
Z' = T' \times _{U \times _ S U} R = T' \times _ T Z.
We see that Z' is an algebraic space, and Z' \to T' is separated and locally quasi-finite by the discussion in the first paragraph of the proof which showed that R is an algebraic space and that the morphism R \to U \times _ S U has those properties. Hence we may apply Lemma 80.5.2 to the diagram
\xymatrix{ Z' \ar[r] \ar[d] & T' \ar[d] \\ Z \ar[r] & T }
and we conclude.
\square
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