Lemma 99.3.4. 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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