Lemma 97.11.2. Let S be a scheme. Let X \to Z \to B and B' \to B be morphisms of algebraic spaces over S. Set Z' = B' \times _ B Z and X' = B' \times _ B X. Then
\text{Res}_{Z'/B'}(X') = B' \times _ B \text{Res}_{Z/B}(X)
in \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}).
Proof.
The equality as functors follows immediately from the definitions. The equality as sheaves follows from this because both sides are sheaves according to Lemma 97.11.1 and the fact that a fibre product of sheaves is the same as the corresponding fibre product of pre-sheaves (i.e., functors).
\square
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