The Stacks project

Proof. We check conditions (1) and (2) of Categories, Definition 4.35.1.

Condition (1). Let $(X \to S, \mathcal{L})$ be an object of $\mathcal{P}\! \mathit{olarized}$ and let $(X' \to S') \to (X \to S)$ be a morphism of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$. Then we let $\mathcal{L}'$ be the pullback of $\mathcal{L}$ to $X'$. Observe that $X, S, S'$ are schemes, hence $X'$ is a scheme as well (as the fibre product of schemes). Then $\mathcal{L}'$ is ample on $X'/S'$ by Morphisms, Lemma 29.37.9. In this way we obtain a morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ lying over $(X' \to S') \to (X \to S)$.

Condition (2). Consider morphisms $(f, g, \varphi ) : (X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ and $(a, b, \psi ) : (Y \to T, \mathcal{N}) \to (X \to S, \mathcal{L})$ of $\mathcal{P}\! \mathit{olarized}$. Given a morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ with $(f, g) \circ (k, h) = (a, b)$ we have to show there is a unique morphism $(k, h, \chi ) : (Y \to T, \mathcal{N}) \to (X' \to S', \mathcal{L}')$ of $\mathcal{P}\! \mathit{olarized}$ such that $(f, g, \varphi ) \circ (k, h, \chi ) = (a, b, \psi )$. We can just take

This proves condition (2). A composition of functors defining fibred categories defines a fibred category, see Categories, Lemma 4.33.12. This we see that $\mathcal{P}\! \mathit{olarized}$ is fibred in groupoids over $\mathit{Sch}_{fppf}$ (strictly speaking we should check the fibre categories are groupoids and apply Categories, Lemma 4.35.2). $\square$