Lemma 79.12.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $X' \subset X$ be the maximal open subspace over which $f$ is locally quasi-finite, see Morphisms of Spaces, Lemma 67.34.7. Then $(X/Y)_{fin} = (X'/Y)_{fin}$.
Proof. Lemma 79.12.2 gives us an injective map $(X'/Y)_{fin} \to (X/Y)_{fin}$. Morphisms of Spaces, Lemma 67.34.7 assures us that formation of $X'$ commutes with base change. Hence everything comes down to proving that if $Z \subset X$ is an open subspace such that $f|_ Z : Z \to Y$ is finite, then $Z \subset X'$. This is true because a finite morphism is locally quasi-finite, see Morphisms of Spaces, Lemma 67.45.8. $\square$
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