The Stacks project

Lemma 65.16.6. Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \to Z$, $h : Y \to Z$ be étale morphisms and let $f : X \to Y$ be a morphism such that $h \circ f = g$. Then $f$ is étale.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_\chi & V \ar[d] \\ X \ar[r] & Y } \]

where $U \to X$ and $V \to Y$ are surjective and étale, see Spaces, Lemma 64.11.6. By assumption the morphisms $\varphi : U \to X \to Z$ and $\psi : V \to Y \to Z$ are étale. Moreover, $\psi \circ \chi = \varphi $ by our assumption on $f, g, h$. Hence $U \to V$ is étale by Lemma 65.16.1 part (2). $\square$

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