The Stacks project

Lemma 65.5.10. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be a presheaf of sets on $(\mathit{Sch}/S)_{fppf}$. The following are equivalent:

  1. the diagonal $F \to F \times F$ is representable,

  2. for $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and any $a \in F(U)$ the map $a : h_ U \to F$ is representable,

  3. for every pair $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and any $a \in F(U)$, $b \in F(V)$ the fibre product $h_ U \times _{a, F, b} h_ V$ is representable.

Proof. This is completely formal, see Categories, Lemma 4.8.4. It depends only on the fact that the category $(\mathit{Sch}/S)_{fppf}$ has products of pairs of objects and fibre products, see Topologies, Lemma 34.7.10. $\square$

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