Remark 99.16.7. In Situation 99.16.3 the rule $(T, g, E) \mapsto (T, g)$ defines a $1$-morphism
\[ \mathcal{C}\! \mathit{omplexes}_{X/B} \longrightarrow \mathcal{S}_ B \]
of stacks in groupoids (see Lemma 99.16.6, 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 $\mathcal{C}\! \mathit{omplexes}_{X'/B'} \to (\mathit{Sch}/S)_{fppf}$ associated to the base change $f' : X' \to B'$. In this situation the diagram
\[ \vcenter { \xymatrix{ \mathcal{C}\! \mathit{omplexes}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{omplexes}_{X/B} \ar[d] \\ \mathcal{S}_{B'} \ar[r] & \mathcal{S}_ B } } \quad \begin{matrix} \text{or in}
\\ \text{another}
\\ \text{notation}
\end{matrix} \quad \vcenter { \xymatrix{ \mathcal{C}\! \mathit{omplexes}_{X'/B'} \ar[r] \ar[d] & \mathcal{C}\! \mathit{omplexes}_{X/B} \ar[d] \\ \mathit{Sch}/B' \ar[r] & \mathit{Sch}/B } } \]
is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space.
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