The Stacks project

Lemma 91.5.2. 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 on objects, 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 $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $\gamma : q(p(x)) \to z_ i|_ U$ be an isomorphism. As $q$ is limit preserving on objects there exist an $i' \geq i$, an object $y_{i'}$ of $\mathcal{Y}$ over $U_{i'}$, an isomorphism $\beta : y_{i'}|_ U \to p(x)$, and an isomorphism $\gamma _{i'} : q(y_{i'}) \to z_ i|_{U_{i'}}$ such that (91.5.0.1) is commutative. As $p$ is limit preserving on objects there exist an $i'' \geq i'$, an object $x_{i''}$ of $\mathcal{X}$ over $U_{i''}$, an isomorphism $\beta ' : x_{i''}|_ U \to x$, and an isomorphism $\gamma '_{i''} : p(x_{i''}) \to y_{i'}|_{U_{i''}}$ such that (91.5.0.1) is commutative. The solution is to take $x_{i''}$ over $U_{i''}$ with isomorphism

\[ q(p(x_{i''})) \xrightarrow {q(\gamma '_{i''})} q(y_{i'})|_{U_{i''}} \xrightarrow {\gamma _{i'}|_{U_{i''}}} z_ i|_{U_{i''}} \]

and isomorphism $\beta ' : x_{i''}|_ U \to x$. We omit the verification that (91.5.0.1) is commutative. $\square$


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