In Sites, Lemma \ref{sites-lemma-bigger-site} we have seen that the functor $f^{-1} : \Sh((\Sch/S)_{fppf}) \to \Sh((\Sch'/S)_{fppf})$ is fully faithful (see discussion in Topologies, Section \ref{topologies-section-change-alpha}). Hence we see that the displayed functor of the lemma is fully faithful. \medskip\noindent Suppose that $X' \in \Ob(\textit{Spaces}'/S)$ such that there exists $U \in \Ob((\Sch/S)_{fppf})$ and a map $U \to X'$ in $\Sh((\Sch'/S)_{fppf})$ which is surjective as a map of sheaves. Let $U' \to X'$ be a surjective \'etale morphism with $U' \in \Ob((\Sch'/S)_{fppf})$. Let $\kappa = \text{size}(U)$, see Sets, Section \ref{sets-section-categories-schemes}. Then $U$ has an affine open covering $U = \bigcup_{i \in I} U_i$ with $|I| \leq \kappa$. Observe that $U' \times_{X'} U \to U$ is \'etale and surjective. For each $i$ we can pick a quasi-compact open $U'_i \subset U'$ such that $U'_i \times_{X'} U_i \to U_i$ is surjective (because the scheme $U' \times_{X'} U_i$ is the union of the Zariski opens $W \times_{X'} U_i$ for $W \subset U'$ affine and because $U' \times_{X'} U_i \to U_i$ is \'etale hence open). Then $\coprod_{i \in I} U'_i \to X$ is surjective \'etale because of our assumption that $U \to X$ and hence $\coprod U_i \to X$ is a surjection of sheaves (details omitted). Because $U'_i \times_{X'} U \to U'_i$ is a surjection of sheaves and because $U'_i$ is quasi-compact, we can find a quasi-compact open $W_i \subset U'_i \times_{X'} U$ such that $W_i \to U'_i$ is surjective as a map of sheaves (details omitted). Then $W_i \to U$ is \'etale and we conclude that $\text{size}(W_i) \leq \text{size}(U)$, see Sets, Lemma \ref{sets-lemma-bound-finite-type}. By Sets, Lemma \ref{sets-lemma-bound-by-covering} we conclude that $\text{size}(U'_i) \leq \text{size}(U)$. Hence $\coprod_{i \in I} U'_i$ is isomorphic to an object of $(\Sch/S)_{fppf}$ by Sets, Lemma \ref{sets-lemma-bound-size}. \medskip\noindent Now let $X'$, $U \to X'$ and $R \to U \times_{X'} U$ be as in the statement of the lemma. In the previous paragraph we have seen that we can find $U' \in \Ob((\Sch/S)_{fppf})$ and a surjective \'etale morphism $U' \to X'$ in $\Sh((\Sch'/S)_{fppf})$. Then $U' \times_{X'} U \to U'$ is a surjection of sheaves, i.e., we can find an fppf covering $\{U'_i \to U'\}$ such that $U'_i \to U'$ factors through $U' \times_{X'} U \to U'$. By Sets, Lemma \ref{sets-lemma-bound-fppf-covering} we can find $\tilde U \to U'$ which is surjective, flat, and locally of finite presentation, with $\text{size}(\tilde U) \leq \text{size}(U')$, such that $\tilde U \to U'$ factors through $U' \times_{X'} U \to U'$. Then we consider $$ \xymatrix{ U' \times_{X'} U' \ar[d] & \tilde U \times_{X'} \tilde U  \ar[l] \ar[d] \ar[r] & U \times_{X'} U \ar[d] \\ U' \times_S U' & \tilde U \times_S \tilde U \ar[l] \ar[r] & U \times_S U } $$ The squares are cartesian. We know the objects of the bottom row are represented by objects of $(\Sch/S)_{fppf}$. By the result of the argument of the previous paragraph, the same is true for $U \times_{X'} U$ (as we have the surjection of sheaves $R \to U \times_{X'} U$ by assumption). Since $(\Sch/S)_{fppf}$ is closed under fibre products (by construction), we see that $\tilde U \times_{X'} \tilde U$ is represented by an object of $(\Sch/S)_{fppf}$. Finally, the map $\tilde U \times_{X'} \tilde U \to U' \times_{X'} U'$ is a surjection of fppf sheaves as $\tilde U \to U'$ is so. Thus we can once more apply the result of the previous paragraph to conclude that $R' = U' \times_{X'} U'$ is represented by an object of $(\Sch/S)_{fppf}$. At this point Lemma \ref{lemma-space-presentation} and Theorem \ref{theorem-presentation} imply that $X = h_{U'}/h_{R'}$ is an object of $\textit{Spaces}/S$ such that $f^{-1}X \cong X'$ as desired.