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
\xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ u \ar[d] \ar[r] & v \ar[d] \\ x \ar[r] & y }
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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