The functor $F_n$ is a sheaf for the fppf topology by Lemma \ref{lemma-flat-dimension-n}. Since $F_n \to Y$ is a monomorphism of sheaves on $(\Sch/S)_{fppf}$ we see that $\Delta : F_n \to F_n \times F_n$ is the pullback of the diagonal $\Delta_Y : Y \to Y \times_S Y$. Hence the representability of $\Delta_Y$ implies the same thing for $F_n$. Therefore it suffices to prove that there exists a scheme $W$ over $S$ and a surjective \'etale morphism $W \to F_n$. \medskip\noindent To construct $W \to F_n$ choose an \'etale covering $\{Y_i \to Y\}$ with $Y_i$ a scheme. Let $X_i = X \times_Y Y_i$ and let $\mathcal{F}_i$ be the pullback of $\mathcal{F}$ to $X_i$. Then $\mathcal{F}_i$ is pure relative to $Y_i$ either by definition or by Lemma \ref{lemma-quasi-finite-base-change}. The other assumptions of the theorem are preserved as well. Finally, the restriction of $F_n$ to $Y_i$ is the functor $F_n$ corresponding to $X_i \to Y_i$ and $\mathcal{F}_i$. Hence it suffices to show: Given $\mathcal{F}$ and $f : X \to Y$ as in the statement of the theorem where $Y$ is a scheme, the functor $F_n$ is representable by a scheme $Z_n$ and $Z_n \to Y$ is a monomorphism of finite presentation. \medskip\noindent Observe that a monomorphism of finite presentation is separated and quasi-finite (Morphisms, Lemma \ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}). Hence combining Descent, Lemma \ref{descent-lemma-descent-data-sheaves}, More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} , and Descent, Lemmas \ref{descent-lemma-descending-property-monomorphism} and \ref{descent-lemma-descending-property-finite-presentation} we see that the question is local for the \'etale topology on $Y$. \medskip\noindent In particular the situation is local for the Zariski topology on $Y$ and we may assume that $Y$ is affine. In this case the dimension of the fibres of $f$ is bounded above, hence we see that $F_n$ is representable for $n$ large enough. Thus we may use descending induction on $n$. Suppose that we know $F_{n + 1}$ is representable by a monomorphism $Z_{n + 1} \to Y$ of finite presentation. Consider the base change $X_{n + 1} = Z_{n + 1} \times_Y X$ and the pullback $\mathcal{F}_{n + 1}$ of $\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \to Y$ is quasi-finite as it is a monomorphism of finite presentation, hence Lemma \ref{lemma-quasi-finite-base-change} implies that $\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$. Since $F_n$ is a subfunctor of $F_{n + 1}$ we conclude that in order to prove the result for $F_n$ it suffices to prove the result for the corresponding functor for the situation $\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$. In this way we reduce to proving the result for $F_n$ in case $Y_{n + 1} = Y$, i.e., we may assume that $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $Y$. \medskip\noindent Fix $n$ and assume $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over the affine scheme $Y$. To finish the proof we have to show that $F_n$ is representable by a monomorphism $Z_n \to S$ of finite presentation. Since the question is local in the \'etale topology on $Y$ it suffices to show that for every $y \in Y$ there exists an \'etale neighbourhood $(Y', y') \to (Y, y)$ such that the result holds after base change to $Y'$. Thus by Lemma \ref{lemma-existence-complete} we may assume there exist \'etale morphisms $h_j : W_j \to X$, $j = 1, \ldots, m$ such that for each $j$ there exists a complete d\'evissage of $\mathcal{F}_j/W_j/Y$ over $y$, where $\mathcal{F}_j$ is the pullback of $\mathcal{F}$ to $W_j$ and such that $|X_y| \subset \bigcup h_j(W_j)$. Since $h_j$ is \'etale, by Lemma \ref{lemma-pre-flat-dimension-n} the sheaves $\mathcal{F}_j$ are still flat over in dimensions $\geq n + 1$ over $Y$. Set $W = \bigcup h_j(W_j)$, which is a quasi-compact open of $X$. As $\mathcal{F}$ is pure along $X_y$ we see that $$ E = \{t \in |Y| : \text{Ass}_{X_t}(\mathcal{F}_t) \subset W \}. $$ contains all generalizations of $y$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relative-assassin-constructible} $E$ is a constructible subset of $Y$. We have seen that $\Spec(\mathcal{O}_{Y, y}) \subset E$. By Morphisms, Lemma \ref{morphisms-lemma-constructible-containing-open} we see that $E$ contains an open neighbourhood of $y$. Hence after shrinking $Y$ we may assume that $E = Y$. It follows from Lemma \ref{lemma-localize-flat-dimension-n} that it suffices to prove the lemma for the functor $F_n$ associated to $X = \coprod W_j$ and $\mathcal{F} = \coprod \mathcal{F}_j$. If $F_{j, n}$ denotes the functor for $W_j \to Y$ and the sheaf $\mathcal{F}_j$ we see that $F_n = \prod F_{j, n}$. Hence it suffices to prove each $F_{j, n}$ is representable by some monomorphism $Z_{j, n} \to Y$ of finite presentation, since then $$ Z_n = Z_{1, n} \times_Y \ldots \times_Y Z_{m, n} $$ Thus we have reduced the theorem to the special case handled in More on Flatness, Lemma \ref{flat-lemma-flat-dimension-n-representable}.