Lemma 32.20.4. Notation and assumptions as in Lemma 32.20.3. Let U \subset U' \subset X be an open containing s.
Let f' : X \to U' correspond to f : X' \to U and g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) via the equivalence. If f and g are separated, proper, finite, étale, then after possibly shrinking U' the morphism f' has the same property.
Let a : X_1 \to X_2 be a morphism of schemes of finite presentation over U' with base change a' : X'_1 \to X'_2 over U and b : Y_1 \to Y_2 over \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}). If a' and b are separated, proper, finite, étale, then after possibly shrinking U' the morphism a has the same property.
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