Lemma 70.3.6. 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 70.3.2. 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
Comments (0)