The Stacks project

Proof. Suppose we have a scheme $T/B$ and two quotients $\mathcal{F}_ T \to \mathcal{Q}_ i$, $i = 1, 2$ corresponding to $T$-valued points of $\mathrm{Quot}_{\mathcal{F}/X/B}$ over $B$. Denote $\mathcal{K}_1$ the kernel of the first one and set $u : \mathcal{K}_1 \to \mathcal{Q}_2$ the composition. By Flatness on Spaces, Lemma 77.8.6 there is a closed subspace of $T$ such that $T' \to T$ factors through it if and only if the pullback $u_{T'}$ is zero. This proves the diagonal is a closed immersion. Moreover, if $\mathcal{F}$ is of finite type, then $\mathcal{K}_1$ is of finite type (Modules on Sites, Lemma 18.24.1) and we see that the diagonal is of finite presentation by the same lemma. $\square$