# Open subfunctor of Quot Functor induced by open immersion

Let $$f: X \rightarrow S$$ be a morphism of noetherian schemes and $$\mathfrak{Q}uot_{\mathcal{E}/X/S}$$ be the functor parametrizing families of quotients of $$\mathcal{E}$$ in the category of locally noetherian schemes over $$S$$. That is, the functor taking any morphism of locally noetherian schemes $$S' \rightarrow S$$ to the set

$$\left\{ (\mathcal{F},q) \> | \> \mathcal{F} \text{ is a coherent sheaf on } X_{S'}, \mathcal{F} \text{ flat over } S', Supp(\mathcal{F}) \text{ proper over } S' \right. \\ \left.\text{ and } q: \mathcal{E}_{S'} \rightarrow \mathcal{F} \text{ surjective } \mathcal{O}_{X_{S'}}\text{-linear morphism} \right\} /\sim$$ where $$\mathcal{E}_{S'}$$ stands for the pullback of $$\mathcal{E}$$ via $$X_{S'} =X \times_S S' \rightarrow X$$ and $$(\mathcal{F},q) \sim (\mathcal{F}',q')$$ if $$\ker(q) = \ker(q')$$.

It is an exercise in Nitin Nitsure's "Construction of Hilbert and Quot schemes" on page 130 of the book "FGA Explained" to show that if $$f: X \rightarrow S$$ is a proper morphism and if $$\iota: U \hookrightarrow X$$ is an open subscheme, then the functor $$\mathfrak{Q}uot_{\mathcal{E}|_U/U/S}$$ is an open subfunctor of $$\mathfrak{Q}uot_{E/X/S}$$.

Question: How to construct the morphism $$\mathfrak{Q}uot_{\mathcal{E}|_U/U/S} \rightarrow \mathfrak{Q}uot_{\mathcal{E}/X/S}$$ when $$\iota$$ is an open embedding?

Remark 1: If $$\iota$$ is a closed immersion, then for each morphism $$S' \rightarrow S$$ the map $$\langle \mathcal{F}, q \rangle \mapsto \langle (\iota_{S'})_*\mathcal{F}, (\iota_{S'})_*q \rangle$$ where $$\iota_{S'}$$ stands for the base change of $$\iota$$ should produce a morphism between the desired functors, however if $$\iota$$ is an open immersion thanks to properness of the support of $$\mathcal{F}$$ one can prove that $$(\iota_{S'})_*\mathcal{F}$$ is coherent, but I don't see how $$(\iota_{S'})_*\mathcal{F}$$ would have proper support over $$S'$$. Maybe there is something I am missing?

Remark 2: Also in the book Nitsure claims that $$\mathfrak{Q}uot_{\mathcal{E}|_U/U/S}$$ is an open subfunctor of $$\mathfrak{Q}uot_{\mathcal{E}/X/S}$$ is a consequence of the following:

Let $$f:X\rightarrow S$$ be a proper morphism of noetherian schemes. Let $$Z \subset X$$ be a closed subscheme, and let $$\mathcal{F}$$ be a coherent sheaf on $$Z$$. Then there exists an open subscheme $$S' \subset S$$ with the universal property that a morphism $$T \rightarrow S$$ factors through $$S'$$ if and only if the support of the pullback $$\mathcal{F}_T$$ on $$Z_T = Z \times_S T$$ is disjoint from $$Y_T = Y \times_S T$$.

It's also unclear to me if this is also used to prove that the morphism $$\mathfrak{Q}uot_{\mathcal{E}|_U/U/S} \rightarrow \mathfrak{Q}uot_{\mathcal{E}/X/S}$$ is well defined in addition to proving it is an open subfunctor.