Lemma 73.20.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is 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 of pairs $a : (U, u) \to (X, x)$, a smooth morphism $b : V \to Y$, and a morphism $h : U \to V$ such that $f \circ a = b \circ h$ and $h$ 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 $a$, $b$ smooth 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 $U \to X \times _ Y V$ smooth 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 a smooth 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 $i \in I$ a smooth 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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