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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