The Stacks project

Lemma 91.5.1. 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 on objects, 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 $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $w$ be an object of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U$, and let $\delta : p'(w) \to z_ i|_ U$ be an isomorphism. We may write $w = (U, x, z, \alpha )$ for some object $x$ of $\mathcal{X}$ over $U$ and object $z$ of $\mathcal{Z}$ over $U$ and isomorphism $\alpha : p(x) \to q(z)$. Note that $p'(w) = z$ hence $\delta : z \to z_ i|_ U$. Set $y_ i = q(z_ i)$ and $\gamma = q(\delta ) \circ \alpha : p(x) \to y_ i|_ U$. As $p$ is limit preserving on objects there exists an $i' \geq i$ and an object $x_{i'}$ of $\mathcal{X}$ over $U_{i'}$ as well as isomorphisms $\beta : x_{i'}|_ U \to x$ and $\gamma _{i'} : p(x_{i'}) \to y_ i|_{U_{i'}}$ such that (91.5.0.1) commutes. Then we consider the object $w_{i'} = (U_{i'}, x_{i'}, z_ i|_{U_{i'}}, \gamma _{i'})$ of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U_{i'}$ and define isomorphisms

\[ w_{i'}|_ U = (U, x_{i'}|_ U, z_ i|_ U, \gamma _{i'}|_ U) \xrightarrow {(\beta , \delta ^{-1})} (U, x, z, \alpha ) = w \]

and

\[ p'(w_{i'}) = z_ i|_{U_{i'}} \xrightarrow {\text{id}} z_ i|_{U_{i'}}. \]

These combine to give a solution to the problem. $\square$


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