Lemma 66.16.6. 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 65.11.6. 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 66.16.1 part (2). \square
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