Lemma 96.3.3. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is limit preserving, then so is the base change $p' : \mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.

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 $S$. For each $i$ we have

$(\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{Z}_{U_ i}$

Filtered colimits commute with $2$-fibre products of categories (details omitted) hence if $p$ is limit preserving we get

\begin{align*} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_{U_ i} & = \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \times _{\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \times _{\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{Z}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = (\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \end{align*}

as desired. $\square$

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