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

$$ \left\{ (\mathcal{F},q) \> | \> \mathcal{F} \text{ is a coherent sheaf on } X_{S'}, Supp(\mathcal{F}) \text{ proper over } S' \right. \\ \left.\text{ and } q: \mathcal{E}_{S'} \rightarrow \mathcal{F} \text{ surjective } \mathcal{O}_{X_{S'}}\text{-linear morphism} \right\} /\sim $$ 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.

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.