Lemma 94.15.1. Let S be a scheme contained in \mathit{Sch}_{fppf}. Let f : \mathcal{X} \to \mathcal{Y} be a 1-morphism of algebraic stacks over S. Let V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}). Let y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y} be surjective and smooth. Then there exists an object U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}) and a 2-commutative diagram
\xymatrix{ (\mathit{Sch}/U)_{fppf} \ar[r]_ a \ar[d]_ x & (\mathit{Sch}/V)_{fppf} \ar[d]^ y \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} }
with x surjective and smooth.
Proof.
First choose W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}) and a surjective smooth 1-morphism z : (\mathit{Sch}/W)_{fppf} \to \mathcal{X}. As \mathcal{Y} is an algebraic stack we may choose an equivalence
j : \mathcal{S}_ F \longrightarrow (\mathit{Sch}/W)_{fppf} \times _{f \circ z, \mathcal{Y}, y} (\mathit{Sch}/V)_{fppf}
where F is an algebraic space. By Lemma 94.10.6 the morphism \mathcal{S}_ F \to (\mathit{Sch}/W)_{fppf} is surjective and smooth as a base change of y. Hence by Lemma 94.10.5 we see that \mathcal{S}_ F \to \mathcal{X} is surjective and smooth. Choose an object U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}) and a surjective étale morphism U \to F. Then applying Lemma 94.10.5 once more we obtain the desired properties.
\square
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