The Stacks project

Lemma 63.5.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $\mathcal{P}$ be a property as in Definition 63.5.1. Let $a : F \to G$ be a representable transformations of functors. Let $b : H \to G$ be any transformation of functors. Consider the fibre product diagram

\[ \xymatrix{ H \times _{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^ a \\ H \ar[r]^ b & G } \]

If $a$ has property $\mathcal{P}$ then also the base change $a'$ has property $\mathcal{P}$.

Proof. Note that the lemma makes sense by Lemma 63.3.3. Proof omitted. $\square$

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