Lemma 81.4.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor
is an equivalence of categories.
Proof. Observe that the base change of a proper surjective morphism is proper and surjective, see Morphisms of Spaces, Lemmas 67.40.3 and 67.5.5. Hence by Lemma 81.4.2 we may work étale locally on $Y$. Hence we reduce to $Y$ being an affine scheme; some details omitted.
Assume $Y$ is affine. By Lemma 81.4.4 it suffices to find a morphism $X' \to X$ where $X'$ is a scheme such that $X' \to Y$ is surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.
In case $X \to Y$ is integral and surjective, we can take $X = X'$ as an integral morphism is representable.
If $f$ is proper and surjective, then the algebraic space $X$ is quasi-compact and separated, see Morphisms of Spaces, Section 67.8 and Lemma 67.4.9. Choose a scheme $X'$ and a surjective finite morphism $X' \to X$, see Limits of Spaces, Proposition 70.16.1. Then $X' \to Y$ is surjective and proper.
Finally, if $X \to Y$ is surjective and flat and locally of finite presentation then we can take an affine étale covering $\{ U_ i \to X\} $ and set $X'$ equal to the disjoint $\coprod U_ i$. $\square$
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