Remark 99.5.5. In Situation 99.5.1 the rule (T, g, \mathcal{F}) \mapsto (T, g) defines a 1-morphism
\mathcal{C}\! \mathit{oh}_{X/B} \longrightarrow \mathcal{S}_ B
of stacks in groupoids (see Lemma 99.5.4, Algebraic Stacks, Section 94.7, and Examples of Stacks, Section 95.10). Let B' \to B be a morphism of algebraic spaces over S. Let \mathcal{S}_{B'} \to \mathcal{S}_ B be the associated 1-morphism of stacks fibred in sets. Set X' = X \times _ B B'. We obtain a stack in groupoids \mathcal{C}\! \mathit{oh}_{X'/B'} \to (\mathit{Sch}/S)_{fppf} associated to the base change f' : X' \to B'. In this situation the diagram
\vcenter { \xymatrix{ \mathcal{C}\! \mathit{oh}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_ B } } \quad \begin{matrix} \text{or in}
\\ \text{another}
\\ \text{notation}
\end{matrix} \quad \vcenter { \xymatrix{ \mathcal{C}\! \mathit{oh}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}_{X/B} \ar[d] \\ \mathit{Sch}/B' \ar[r] & \mathit{Sch}/B } }
is 2-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.
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