Lemma 94.19.1. Let \mathit{Sch}_{fppf} be a big fppf site. Let S \to S' be a morphism of this site. The constructions A and B of Stacks, Section 8.13 above give isomorphisms of 2-categories
\left\{ \begin{matrix} 2\text{-category of algebraic}
\\ \text{stacks }\mathcal{X}\text{ over }S
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} 2\text{-category of pairs }(\mathcal{X}', f)\text{ consisting of an}
\\ \text{algebraic stack }\mathcal{X}'\text{ over }S'\text{ and a morphism}
\\ f : \mathcal{X}' \to (\mathit{Sch}/S)_{fppf}\text{ of algebraic stacks over }S'
\end{matrix} \right\}
Proof. The statement makes sense as the functor j : (\mathit{Sch}/S)_{fppf} \to (\mathit{Sch}/S')_{fppf} is the localization functor associated to the object S/S' of (\mathit{Sch}/S')_{fppf}. By Stacks, Lemma 8.13.2 the only thing to show is that the constructions A and B preserve the subcategories of algebraic stacks. For example, if \mathcal{X} = [U/R] then construction A applied to \mathcal{X} just produces \mathcal{X}' = \mathcal{X}. Conversely, if \mathcal{X}' = [U'/R'] the morphism p induces morphisms of algebraic spaces U' \to S and R' \to S, and then \mathcal{X} = [U'/R'] but now viewed as a stack over S. Hence the lemma is clear. \square
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Comment #193 by Pieter Belmans on
Comment #200 by Johan on