The Stacks project

Situation 77.7.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $u : \mathcal{F} \to \mathcal{G}$ be a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $B$ we will denote $u_ T : \mathcal{F}_ T \to \mathcal{G}_ T$ the base change of $u$ to $T$, in other words, $u_ T$ is the pullback of $u$ via the projection morphism $X_ T = X \times _ B T \to X$. In this situation we can consider the functor

There are variants $F_{inj}$, $F_{surj}$, $F_{zero}$ where we ask that $u_ T$ is injective, surjective, or zero.