The Stacks project

Proof. Set $\mathcal{G} = \pi _*\mathcal{F}$. Since $\pi $ is affine we see that for any scheme $T$ over $B$ we have $\mathcal{G}_ T = \pi _{T, *}\mathcal{F}_ T$ by Cohomology of Spaces, Lemma 69.11.1. Moreover $\pi _ T$ is affine, hence $\pi _{T, *}$ is exact and transforms quotients into quotients. Observe that a quasi-coherent quotient $\mathcal{F}_ T \to \mathcal{Q}$ defines a point of $\mathrm{Quot}_{X/B}$ if and only if $\mathcal{Q}$ defines an object of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $T$ (similarly for $\mathcal{G}$ and $Y$). Since we've seen in Lemma 108.4.4 that $\pi _*$ induces a morphism $\mathcal{C}\! \mathit{oh}_{X/B} \to \mathcal{C}\! \mathit{oh}_{Y/B}$ we see that if $\mathcal{F}_ T \to \mathcal{Q}$ is in $\mathrm{Quot}_{\mathcal{F}/X/B}(T)$, then $\mathcal{G}_ T \to \pi _{T, *}\mathcal{Q}$ is in $\mathrm{Quot}_{\mathcal{G}/Y/B}(T)$. $\square$