Lemma 102.3.4. Let p : \mathcal{X} \to \mathcal{Y} and q : \mathcal{Y} \to \mathcal{Z} be 1-morphisms of categories fibred in groupoids over (\mathit{Sch}/S)_{fppf}. If p and q are limit preserving, then so is the composition q \circ p.
Proof. This is formal. Let U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i be the directed limit of affine schemes U_ i over S. If p and q are limit preserving we get
\begin{align*} \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{Y}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \end{align*}
as desired. \square
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