The Stacks project

Remark 99.3.4. In Situation 99.3.1 let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times _ B B'$ and denote $\mathcal{F}'$, $\mathcal{G}'$ the pullback of $\mathcal{F}$, $\mathcal{G}$ to $X'$. Then we obtain a functor $\mathit{Hom}(\mathcal{F}', \mathcal{G}') : (\mathit{Sch}/B')^{opp} \to \textit{Sets}$ associated to the base change $f' : X' \to B'$. For a scheme $T$ over $B'$ it is clear that we have

\[ \mathit{Hom}(\mathcal{F}', \mathcal{G}')(T) = \mathit{Hom}(\mathcal{F}, \mathcal{G})(T) \]

where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. This trivial remark will occasionally be useful to change the base algebraic space.

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