The Stacks project

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

\[ \mathop{\mathrm{colim}}\nolimits (F \times _ G H)(T_ i) = \mathop{\mathrm{colim}}\nolimits F(T_ i) \times _{\mathop{\mathrm{colim}}\nolimits G(T_ i)} \mathop{\mathrm{colim}}\nolimits H(T_ i) \]

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

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits F(T_ i) \times _{\mathop{\mathrm{colim}}\nolimits G(T_ i)} \mathop{\mathrm{colim}}\nolimits H(T_ i) \ar[r] \ar[d] & F(T) \times _{G(T)} H(T) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _ i E(T_ i) \ar[r] & E(T) } \]

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

\[ E' \times _ E (F \times _ G H) = (E' \times _ E F) \times _{(E' \times _ E G)} (E' \times _ E H) \]

Some details omitted. $\square$


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