Lemma 59.29.4. Let S be a scheme, and let \overline{s} be a geometric point of S. The category of étale neighborhoods is cofiltered. More precisely:
Let (U_ i, \overline{u}_ i)_{i = 1, 2} be two étale neighborhoods of \overline{s} in S. Then there exists a third étale neighborhood (U, \overline{u}) and morphisms (U, \overline{u}) \to (U_ i, \overline{u}_ i), i = 1, 2.
Let h_1, h_2: (U, \overline{u}) \to (U', \overline{u}') be two morphisms between étale neighborhoods of \overline{s}. Then there exist an étale neighborhood (U'', \overline{u}'') and a morphism h : (U'', \overline{u}'') \to (U, \overline{u}) which equalizes h_1 and h_2, i.e., such that h_1 \circ h = h_2 \circ h.
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