Lemma 67.22.5. Let $\mathcal{Q}$ be a property of morphisms of germs which is étale local on the source-and-target. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ be a point of $X$. Consider the diagrams

where $U$ and $V$ are schemes, $a, b$ are étale, and $u, v, x, y$ are points of the corresponding spaces. The following are equivalent

for any diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$, and

for some diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$.

If $X$ and $Y$ are representable, then this is also equivalent to $\mathcal{Q}((X, x) \to (Y, y))$.

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