The Stacks project

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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