The Stacks project

Proof. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $t, s : R \to U$.

Let $v : T \to \text{Q}_{\mathcal{F}/X/B}$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma 65.9.1), we obtain a morphism $h : T \to B$ such that $h_ U = h \circ p$. By Descent on Spaces, Proposition 74.4.1 the quotient $\mathcal{Q}_ U$ descends to a quotient $\mathcal{F}_ T \to \mathcal{Q}$ over $X_ T$. Since $U \to T$ is surjective and flat, it follows from Morphisms of Spaces, Lemma 67.31.5 that $\mathcal{Q}$ is flat over $T$.

Conversely, let $(h, \mathcal{F}_ T \to \mathcal{Q})$ be a pair over $T$. Then we get a natural transformation $v : T \to \text{Q}_{\mathcal{F}/X/B}$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, \mathcal{F}_{T'} \to a^*\mathcal{Q})$. We omit the verification that the construction of this and the previous paragraph are mutually inverse.

In the case of $\text{Q}^{fp}_{\mathcal{F}/X/B}$ we add: given a morphism $h : T \to B$, a quasi-coherent sheaf on $X_ T$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module if and only if the pullback to $X_ U$ is of finite presentation as an $\mathcal{O}_{X_ U}$-module. This follows from the fact that $X_ U \to X_ T$ is surjective and étale and Descent on Spaces, Lemma 74.6.2. $\square$