The Stacks project

Situation 38.20.11. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. For any scheme $T$ over $S$ we will denote $\mathcal{F}_ T$ the base change of $\mathcal{F}$ to $T$, in other words, $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ via the projection morphism $X_ T = X \times _ S T \to X$. Note that $X_ T \to T$ is of finite type and that $\mathcal{F}_ T$ is an $\mathcal{O}_{X_ T}$-module of finite type (Morphisms, Lemma 29.15.4 and Modules, Lemma 17.9.2). Let $n \geq 0$. By Definition 38.20.10 and Lemma 38.20.9 we obtain a functor