Lemma 79.6.1. Let $S$ be a scheme. Let $X \to X'$ be a thickening of schemes over $S$ and let $X \to Y$ be an affine morphism of schemes over $S$. Let $Y' = Y \amalg _ X X'$ be the pushout in the category of schemes (see More on Morphisms, Lemma 37.14.3). Then $Y'$ is also a pushout in the category of algebraic spaces over $S$.

**Proof.**
This is an immediate consequence of Lemma 79.3.1 and More on Morphisms, Lemmas 37.14.3, 37.14.4, and 37.14.6.
$\square$

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