Lemma 70.3.4. 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 b \circ a and b are limit preserving, then a 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 lower square and the outer rectangle are fibre products of sets. Hence the upper square is a fibre product square too which proves the lemma. \square
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