Lemma 37.34.3. Let $S$ be a scheme, and let $s$ be a point of $S$. The category of étale neighborhoods has the following properties:

Let $(U_ i, u_ i)_{i=1, 2}$ be two étale neighborhoods of $s$ in $S$. Then there exists a third étale neighborhood $(U, u)$ and morphisms $(U, u) \to (U_ i, u_ i)$, $i = 1, 2$.

Let $h_1, h_2: (U, u) \to (U', u')$ be two morphisms between étale neighborhoods of $s$. Assume $h_1$, $h_2$ induce the same map $\kappa (u') \to \kappa (u)$ of residue fields. Then there exist an étale neighborhood $(U'', u'')$ and a morphism $h : (U'', u'') \to (U, u)$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$.

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