Lemma 69.3.6. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $E, F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$, and $c : G \to E$ be transformations of functors. If $c$, $c \circ a$, and $c \circ b$ are limit preserving, then $F \times _ G H \to E$ is too.

**Proof.**
Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in characterization (2) of Lemma 69.3.2. Then we have

as filtered colimits commute with finite products. Our goal is thus to show that

is a fibre product diagram. This follows from the observation that given maps of sets $E' \to E$, $F \to G$, $H \to G$, and $G \to E$ we have

Some details omitted. $\square$

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