Lemma 73.21.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is etale-smooth local on source-and-target. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

$f$ has property $\mathcal{P}$,

for every $x \in |X|$ there exists a smooth morphism $b : V \to Y$, an étale morphism $a : U \to V \times _ Y X$, and a point $u \in |U|$ mapping to $x$ such that $U \to V$ has $\mathcal{P}$,

for some commutative diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]with $b$ smooth, $U \to V \times _ Y X$ étale, and $a$ surjective the morphism $h$ has $\mathcal{P}$,

for any commutative diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]with $b$ smooth and $U \to X \times _ Y V$ étale, the morphism $h$ has $\mathcal{P}$,

there exists a smooth covering $\{ Y_ i \to Y\} _{i \in I}$ such that each base change $Y_ i \times _ Y X \to Y_ i$ has $\mathcal{P}$,

there exists an étale covering $\{ X_ i \to X\} _{i \in I}$ such that each composition $X_ i \to Y$ has $\mathcal{P}$,

there exists a smooth covering $\{ Y_ i \to Y\} _{i \in I}$ and for each $i \in I$ an étale covering $\{ X_{ij} \to Y_ i \times _ Y X\} _{j \in J_ i}$ such that each morphism $X_{ij} \to Y_ i$ has $\mathcal{P}$.

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