Lemma 76.9.6. Let S be a scheme. Let X \subset X' be a thickening of algebraic spaces over S. The functor
V' \longmapsto V = X \times _{X'} V'
defines an equivalence of categories X'_{\acute{e}tale}\to X_{\acute{e}tale}.
Proof. The functor V' \mapsto V defines an equivalence of categories X'_{spaces, {\acute{e}tale}} \to X_{spaces, {\acute{e}tale}}, see Theorem 76.8.1. Thus it suffices to show that V is a scheme if and only if V' is a scheme. This is the content of Lemma 76.9.5. \square
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