Lemma 76.15.3. Let $S$ be a scheme. Let $Z \to Y \to X$ be morphisms of algebraic spaces over $S$. Assume $Z \to Y$ is étale.

If $Y \subset Y'$ is a universal first order thickening of $Y$ over $X$, then the unique étale morphism $Z' \to Y'$ such that $Z = Y \times _{Y'} Z'$ (see Theorem 76.8.1) is a universal first order thickening of $Z$ over $X$.

If $Z \to Y$ is surjective and $(Z \subset Z') \to (Y \subset Y')$ is an étale morphism of first order thickenings over $X$ and $Z'$ is a universal first order thickening of $Z$ over $X$, then $Y'$ is a universal first order thickening of $Y$ over $X$.

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