Lemma 97.5.6. Let S be a scheme. Let f : \mathcal{X} \to \mathcal{Y} be a 1-morphism of categories fibred in groupoids over (\mathit{Sch}/S)_{fppf}. Let \mathcal{P} be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition 94.10.1. If
f is representable by algebraic spaces,
\mathcal{Y} \to (\mathit{Sch}/S)_{fppf} is limit preserving on objects,
for an affine scheme V locally of finite presentation over S and y \in \mathcal{Y}_ V the resulting morphism of algebraic spaces f_ y : F_ y \to V, see Algebraic Stacks, Equation (94.9.1.1), has property \mathcal{P}.
Then f has property \mathcal{P}.
Proof.
Let V be a scheme over S and y \in \mathcal{Y}_ V. We have to show that F_ y \to V has property \mathcal{P}. Since \mathcal{P} is fppf local on the base we may assume that V is an affine scheme which maps into an affine open \mathop{\mathrm{Spec}}(\Lambda ) \subset S. Thus we can write V = \mathop{\mathrm{lim}}\nolimits V_ i with each V_ i affine and of finite presentation over \mathop{\mathrm{Spec}}(\Lambda ), see Algebra, Lemma 10.127.2. Then y comes from an object y_ i over V_ i for some i by assumption (2). By assumption (3) the morphism F_{y_ i} \to V_ i has property \mathcal{P}. As \mathcal{P} is stable under arbitrary base change and since F_ y = F_{y_ i} \times _{V_ i} V we conclude that F_ y \to V has property \mathcal{P} as desired.
\square
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