Assume (1). By Limits of Spaces, Lemma \ref{spaces-limits-lemma-locally-finite-presentation-permanence} the transformation $a'$ is limit preserving\footnote{This makes sense even if $\tau \not = fppf$ as the underlying category of $(\Sch/S)_\tau$ equals the underlying category of $(\Sch/S)_{fppf}$ and the statement doesn't refer to the topology.}. Take $\xi : V \to G$ as in (2). Denote $V' = V$ but viewed as an object of $(\Sch/S)_\tau$. Since $G$ is the restriction of $G'$ to $(\textit{Noetherian}/S)_\tau$ we see that $\xi \in G(V)$ corresponds to $\xi' \in G'(V')$. By assumption $V' \times_{\xi', G'} F'$ is representable by a scheme $U'$. The morphism of schemes $U' \to V'$ corresponding to the projection $V' \times_{\xi', G'} F' \to V'$ is locally of finite presentation by Limits of Spaces, Lemma \ref{spaces-limits-lemma-base-change-locally-finite-presentation} and Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation}. Hence $U'$ is a locally Noetherian scheme and therefore $U'$ is isomorphic to an object $U$ of $(\textit{Noetherian}/S)_\tau$. Then $U$ represents $F \times_G V$ as desired. \medskip\noindent The implication (2) $\Rightarrow$ (3) is immediate. Assume (3). We will prove (1). Let $T$ be an object of $(\Sch/S)_\tau$ and let $T \to G'$ be a morphism. We have to show the functor $F' \times_{G'} T$ is representable by a scheme $X$ over $T$. Let $\mathcal{B}$ be the set of affine opens of $T$ which map into an affine open of $S$. This is a basis for the topology of $T$. Below we will show that for $W \in \mathcal{B}$ the fibre product $F' \times_{G'} W$ is representable by a scheme $X_W$ over $W$. If $W_1 \subset W_2$ in $\mathcal{B}$, then we obtain an isomorphism $X_{W_1} \to X_{W_2} \times_{W_2} W_1$ because both $X_{W_1}$ and $X_{W_2} \times_{W_2} W_1$ represent the functor $F' \times_{G'} W_1$. These isomorphisms are canonical and satisfy the cocycle condition mentioned in Constructions, Lemma \ref{constructions-lemma-relative-glueing}. Hence we can glue the schemes $X_W$ to a scheme $X$ over $T$. Compatibility of the glueing maps with the maps $X_W \to F'$ provide us with a map $X \to F'$. The resulting map $X \to F' \times_{G'} T$ is an isomorphism as we may check this locally on $T$ (as source and target of this arrow are sheaves for the Zariski topology). \medskip\noindent Let $W$ be an affine scheme which maps into an affine open $U \subset S$. Let $W \to G'$ be a map. Still assuming (3) we have to show that $F' \times_{G'} W$ is representable by a scheme. We may write $W = \lim V'_i$ as a directed limit of affine schemes $V'_i$ of finite presentation over $U$, see Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}. Since $V'_i$ is of finite type over an Noetherian scheme, we see that $V'_i$ is a Noetherian scheme. Denote $V_i = V'_i$ but viewed as an object of $(\textit{Noetherian}/S)_\tau$. As $G'$ is limit preserving can choose an $i$ and a map $V'_i \to G'$ such that $W \to G'$ is the composition $W \to V'_i \to G'$. Since $G$ is the restriction of $G'$ to $(\textit{Noetherian}/S)_\tau$ the morphism $V'_i \to G'$ is the same thing as a morphism $V_i \to G$ (see above). By assumption (3) the functor $F \times_G V_i$ is representable by an object $X_i$ of $(\textit{Noetherian}/S)_\tau$. The functor $F \times_G V_i$ is limit preserving as it is the restriction of $F' \times_{G'} V'_i$ and this functor is limit preserving by Limits of Spaces, Lemma \ref{spaces-limits-lemma-fibre-product-locally-finite-presentation}, the assumption that $F'$ and $G'$ are limit preserving, and Limits, Remark \ref{limits-remark-limit-preserving} which tells us that the functor of points of $V'_i$ is limit preserving. By Lemma \ref{lemma-representable-limit-preserving} we conclude that $X_i$ is locally of finite presentation over $S$. Denote $X'_i = X_i$ but viewed as an object of $(\Sch/S)_\tau$. Then we see that $F' \times_{G'} V'_i$ and the functors of points $h_{X'_i}$ are both extensions of $h_{X_i} : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}$ to limit preserving sheaves on $(\Sch/S)_\tau$. By the equivalence of categories of Lemma \ref{lemma-canonical-extension} we deduce that $X'_i$ represents $F' \times_{G'} V'_i$. Then finally $$ F' \times_{G'} W = F' \times_{G'} V'_i \times_{V'_i} W = X'_i \times_{V'_i} W $$ is representable as desired.