Lemma 70.3.3. Let S be a scheme contained in \mathit{Sch}_{fppf}. Let F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}. Let a : F \to G, b : G \to H be transformations of functors. If a and b are limit preserving, then
b \circ a : F \longrightarrow H
is limit preserving.
Proof.
Let T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i as in characterization (2) of Lemma 70.3.2. Consider the diagram of sets
\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i F(T_ i) \ar[r] \ar[d]_ a & F(T) \ar[d]^ a \\ \mathop{\mathrm{colim}}\nolimits _ i G(T_ i) \ar[r] \ar[d]_ b & G(T) \ar[d]^ b \\ \mathop{\mathrm{colim}}\nolimits _ i H(T_ i) \ar[r] & H(T) }
By assumption the two squares are fibre product squares. Hence the outer rectangle is a fibre product diagram too which proves the lemma.
\square
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