The Stacks project

Proof. By More on Morphisms, Lemma 37.14.6 there is an equivalence

where $\textit{flat-lfp}_ S$ signifies the category of schemes flat and locally of finite presentation over $S$. Let $X'/S'$ on the left hand side correspond to the triple $(X/S, Y'/T', \varphi )$ on the right hand side. Set $Y = T \times _{T'} Y'$ which is isomorphic with $T \times _ S X$ via $\varphi $. Then More on Morphisms, Lemma 37.14.5 shows that we have an equivalence

where $\textit{QCoh-flat}_{X/S}$ signifies the category of quasi-coherent $\mathcal{O}_ X$-modules flat over $S$. Since $X \to S$, $Y \to T$, $X' \to S'$, $Y' \to T'$ are flat, this will in particular apply to invertible modules to give an equivalence of categories

where $\textit{Pic}(X)$ signifies the category of invertible $\mathcal{O}_ X$-modules. There is a small point here: one has to show that if an object $\mathcal{F}'$ of $\textit{QCoh-flat}_{X'/S'}$ pulls back to invertible modules on $X$ and $Y'$, then $\mathcal{F}'$ is an invertible $\mathcal{O}_{X'}$-module. It follows from the cited lemma that $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation. By More on Morphisms, Lemma 37.16.7 it suffices to check the restriction of $\mathcal{F}'$ to fibres of $X' \to S'$ is invertible. But the fibres of $X' \to S'$ are the same as the fibres of $X \to S$ and hence these restrictions are invertible.

Having said the above we obtain an equivalence of categories if we drop the assumption (for the category of objects over $S$) that $X \to S$ be proper and the assumption that $\mathcal{L}$ be ample. Now it is clear that if $X' \to S'$ is proper, then $X \to S$ and $Y' \to T'$ are proper (Morphisms, Lemma 29.41.5). Conversely, if $X \to S$ and $Y' \to T'$ are proper, then $X' \to S'$ is proper by More on Morphisms, Lemma 37.3.3. Similarly, if $\mathcal{L}'$ is ample on $X'/S'$, then $\mathcal{L}'|_ X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$ (Morphisms, Lemma 29.37.9). Finally, if $\mathcal{L}'|_ X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$, then $\mathcal{L}'$ is ample on $X'/S'$ by More on Morphisms, Lemma 37.3.2. $\square$