The Stacks project

Lemma 96.7.2. 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 surjective on objects, then so is the composition $q \circ p$.

Proof. This is formal. Let $z$ be an object of $\mathcal{Z}$ over a field $k$. As $q$ is surjective on objects there exists a field extension $K/k$ and an object $y$ of $\mathcal{Y}$ over $K$ such that $q(y) \cong x|_{\mathop{\mathrm{Spec}}(K)}$. As $p$ is surjective on objects there exists a field extension $L/K$ and an object $x$ of $\mathcal{X}$ over $L$ such that $p(x) \cong y|_{\mathop{\mathrm{Spec}}(L)}$. Then the field extension $L/k$ and the object $x$ of $\mathcal{X}$ over $L$ satisfy $q(p(x)) \cong z|_{\mathop{\mathrm{Spec}}(L)}$ as desired. $\square$


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