The Stacks project

Situation 99.3.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $B$ we will denote $\mathcal{F}_ T$ and $\mathcal{G}_ T$ the base changes of $\mathcal{F}$ and $\mathcal{G}$ to $T$, in other words, the pullbacks via the projection morphism $X_ T = X \times _ B T \to X$. We consider the functor