The Stacks project

Situation 99.8.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. For any scheme $T$ over $B$ we will denote $X_ T$ the base change of $X$ to $T$ and $\mathcal{F}_ T$ the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Given such a $T$ we set

By Derived Categories of Spaces, Lemma 75.7.8 this is a subfunctor of the functor $Q^{fp}_{\mathcal{F}/X/B}$ we discussed in Section 99.7. Thus we obtain a functor

This is the Quot functor associated to $\mathcal{F}/X/B$.