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 $p'(w) = z|_{\mathop{\mathrm{Spec}}(K)}$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).