How to Prove Zariski Open Covering for Representable Functors in Algebraic Ge...

Problem

Possibility of being Zariski open covering of a functor for some family I am trying to show reductibility to affine case in showing representability of some kind of functor and came across next question. Next are prelimanery definitions ( C.f. Gortz, Wedhorn's A.G. book ( 8.2 ) ). Let $X$ be a scheme and let $g: X\to G$ be a morphism in $\widehat{(Sch)}$ ( recall that we do not distinguish explicitly between a scheme $X$ and the attached functor $h_X$ ). Let $F \times_G X$ be the fiber product in the category $\widehat{(Sch)}$ ( Example 8.3 ; $(F \times_G X) ( T) := F(T) \times_{G(T)} X(T) $ ). Denote by $p : F \times_G X \to F$, $q : F \times_GX \to X$ the two projections. Definition 8.5. A morphism $f: F\to G$ of functors in $\widehat{(Sch)}$ is called representable if for all schemes $X$ and all morphisms $g: X \to G$ in $\widehat{(Sch)}$ the functor $F\times_G X $ is representable. Let $Z$ be a scheme and let $\zeta : Z \xrightarrow{\cong} F \times_G X$ be an isomorphim. By the Yoneda lemma 4.6 the composition of $\zeta$ with the second projection $F \times_G X \to X$ is given by a unique scheme morphism $Z \to X$ which is independent of the choice of $(Z, \zeta)$ up to composition with an isomorphism. Thus the following definition makes sense. Definition 8.6. Let $\mathbf{P}$ be a property of morphisms of schemes such that the composition of a morphism possessing $\mathbf{P}$ with an isomorphism from the right or from the left has again property $\mathbf{P}$. We say that a representable morphism $f: F \to G$ of functors in $\widehat{(Sch)}$ possesses the property $\mathbf{P}$ if for all schemes $X$ and for every morphism $g : X \to G$ the second projection $f_{(X)} : F\times_G X \to X$ possesses $\mathbf{P}$. Definition ( open subfunctor, Zariski open coverings). Let $S$ be a fixed scheme and let $F : ( \mathrm{Sch}/S )^{\mathrm{opp}} \to (\mathrm{Sets})$ be a contravariant functor. An open subfunctor $F'$ of $F$ is a representable morphism $f: F' \to F$ that is an open immersion ( Definition 8.6). By definition this means that for every $S$-scheme $X$ and for every $S$-morphism $g: X \to F$ the second projection $f_{(X)} : F' \times_F X \to X$ is an open immersion of schemes. A family $( f_i : F_i \to F ) _{ i\in I}$ of open subfunctors is called a Zariski open covering of $F$, if for every $S$-scheme $X$ and every $S$-morphism $g:X \to F$ the images of the $(f_i)_{(X)} : F_i \times_F X \to X$ form a covering of $X$. Let me explain what covering of $X$ means as follows. Since each $f_i$ is representable, $F_i \times_F X$ is a representable functor so that there are isomorphisms $\zeta_i : Z_i \xrightarrow{\cong} F_i \times_F X$. By composing with the $(f_i)_{(X)}$, we have scheme morphisms $l_i : Z_i \to X$. Then covering of $X$ means $\bigcup_{i\in I} l_i(Z_i) =X$. Definition ( Zariski sheaves ). Let $F : ( \mathrm{Sch}/S)^{\mathrm{opp}} \to ( \mathrm{(Sets)}$ be a functor. If $j: U \to X$ is an open immersion of $S$-schemes and $\xi \in F(X)$, we write, like for sections of presheaves, simply $\xi|_U$ instead of $F(j)(\xi)$. We say that $F$ is a Zariski sheaf ( on $(\mathrm{Sch}/S)$) if the usual sheaf axioms are satisfied, that is, for every $S$-scheme $X$ and for every open covering $X= \bigcup_{i \in I}U_i$ we have : Given $\xi_i \in F(U_i) $ for all $i\in I$ such that $\xi_{i}|_{(U_i \cap U_j)} = \xi_{j}|_{(U_i \cap U_j)}$ for all $i,j \in I$, there exists a unique element $\xi \in F(X)$ such that $\xi|_{U_i} =\xi_i$ for all $i\in I$. Q. My question is, let $S$ be a fixed scheme and let $F : ( \mathrm{Sch}/S )^{\mathrm{opp}} \to (\mathrm{Sets})$ be a Zariski sheaf with $S = \bigcup_{i\in I} U_i$ an affine open covering. Then $(p_i : F|_{U_i} := F \times_S U_i \to F )_{i\in I}$ foms a Zariski open covering of $F$? Here $p_i$ are first projections. Unwinding definitions, next are true? 1) The fiber product $F|_{U_i} := F \times_S U_i$ is well-defined? ; i.e., does there a natural morphism $F \to S$? 2) Each $p_i$ is open subfuntor ; i.e., $p_i$ is representable ( I can't show this at all ) and open immersion ? Similarly as an open immersion of schemes is stable under base change, we can try to consider that $p_i$ maybe open subfunctor. But is it really works? 3) The family $(p_i)_{i\in I}$ really forms a Zariski open covering of $F$? Above definitions are somewhat complex to grasp.