The Stacks project

Remark 99.13.5. Let $B$ be an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $B\textit{-Spaces}'_{ft}$ be the category consisting of pairs $(X \to S, h : S \to B)$ where $X \to S$ is an object of $\mathcal{S}\! \mathit{paces}'_{ft}$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', h') \to (X \to S, h)$ in $B\textit{-Spaces}'_{ft}$ is a morphism $(f, g)$ in $\mathcal{S}\! \mathit{paces}'_{ft}$ such that $h \circ g = h'$. In this situation the diagram

\[ \xymatrix{ B\textit{-Spaces}'_{ft} \ar[r] \ar[d] & \mathcal{S}\! \mathit{paces}'_{ft} \ar[d] \\ (\mathit{Sch}/B)_{fppf} \ar[r] & \mathit{Sch}_{fppf} } \]

is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\mathcal{S}\! \mathit{paces}'_{ft}$ to the case of families over a given base algebraic space. Of course, a similar construction works for $B\textit{-Spaces}'_{fp, flat, proper}$

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