Lemma 96.7.1. 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 surjective 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 $z$ be an object of $\mathcal{Z}$ over a field $k$. As $p$ is surjective on objects there exists an extension $K/k$ and an object $x$ of $\mathcal{X}$ over $K$ and an isomorphism $\alpha : p(x) \to q(z)|_{\mathop{\mathrm{Spec}}(K)}$. Then $w = (\mathop{\mathrm{Spec}}(K), x, z|_{\mathop{\mathrm{Spec}}(K)}, \alpha )$ is an object of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $K$ with $p'(w) = z|_{\mathop{\mathrm{Spec}}(K)}$.
$\square$

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