Lemma 81.13.3. In Situation 81.13.2 let $Y = X' \amalg Z$ and $R = Y \times _ X Y$ with projections $t, s : R \to Y$. There exists a coequalizer $X_1$ of $s, t : R \to Y$ in the category of algebraic spaces over $S$. The morphism $X_1 \to X$ is a finite universal homeomorphism, an isomorphism over $U$, and $Z \to X$ lifts to $X_1$.
Proof. Existence of $X_1$ and the fact that $X_1 \to X$ is a finite universal homeomorphism is a special case of Lemma 81.13.1. The formation of $X_1$ commutes with étale localization on $X$ (see proof of Lemma 81.13.1). Thus the morphism $X_1 \to X$ is an isomorphism over $U$. It is immediate from the construction that $Z \to X$ lifts to $X_1$. $\square$
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