Since $j : R \to U \times_S U$ is a monomorphism we see that $j$ is separated (see Schemes, Lemma \ref{schemes-lemma-monomorphism-separated}). Since $U$ is affine we see that $U \times_S U$ (which comes equipped with a monomorphism into the affine scheme $U \times U$) is separated. Hence we see that $R$ is separated. In particular the morphisms $s, t$ are separated as well as \'etale. \medskip\noindent Since the composition $R \to U \times_S U \to U$ is locally of finite type we conclude that $j$ is locally of finite type (see Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type}). As $j$ is also a monomorphism it has finite fibres and we see that $j$ is locally quasi-finite by Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}. Altogether we see that $j$ is separated and locally quasi-finite. \medskip\noindent Our first step is to show that the quotient map $c : U \to F$ is representable. Consider a scheme $T$ and a morphism $a : T \to F$. We have to show that the sheaf $G = T \times_{a, F, c} U$ is representable. As seen in the proofs of Lemmas \ref{lemma-finding-opens} and \ref{lemma-when-it-works-it-works} there exists an fppf covering $\{\varphi_i : T_i \to T\}_{i \in I}$ and morphisms $a_i : T_i \to U$ such that $a_i \times a_{i'} : T_i \times_T T_{i'} \to U \times_S U$ factors through $R$, and such that $c \circ a_i = a \circ \varphi_i$. As in the proof of Lemma \ref{lemma-when-it-works-it-works} we see that \begin{eqnarray*} T_i \times_{\varphi_i, T} G & = & T_i \times_{\varphi_i, T} T \times_{a, U/R, c} U \\ & = & T_i \times_{c \circ a_i, U/R, c} U \\ & = & T_i \times_{a_i, U} U \times_{c, U/R, c} U \\ & = & T_i \times_{a_i, U, t} R \end{eqnarray*} Since $t$ is separated and \'etale, and in particular separated and locally quasi-finite (by Morphisms, Lemmas \ref{morphisms-lemma-unramified-quasi-finite} and \ref{morphisms-lemma-flat-unramified-etale}) we see that the restriction of $G$ to each $T_i$ is representable by a morphism of schemes $X_i \to T_i$ which is separated and locally quasi-finite. By Descent, Lemma \ref{descent-lemma-descent-data-sheaves} we obtain a descent datum $(X_i, \varphi_{ii'})$ relative to the fppf-covering $\{T_i \to T\}$. Since each $X_i \to T_i$ is separated and locally quasi-finite we see by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} that this descent datum is effective. Hence by Descent, Lemma \ref{descent-lemma-descent-data-sheaves} (2) we conclude that $G$ is representable as desired. \medskip\noindent The second step of the proof is to show that $U \to F$ is surjective and \'etale. This is clear from the above since in the first step above we saw that $G = T \times_{a, F, c} U$ is a scheme over $T$ which base changes to schemes $X_i \to T_i$ which are surjective and \'etale. Thus $G \to T$ is surjective and \'etale (see Remark \ref{remark-list-properties-fpqc-local-base}). Alternatively one can reread the proof of Lemma \ref{lemma-when-it-works-it-works} in the current situation. \medskip\noindent The third and final step is to show that the diagonal map $F \to F \times F$ is representable. We first observe that the diagram $$ \xymatrix{ R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\ U \times_S U \ar[r] & F \times F } $$ is a fibre product square. By Lemma \ref{lemma-product-representable-transformations} the morphism $U \times_S U \to F \times F$ is representable (note that $h_U \times h_U = h_{U \times_S U}$). Moreover, by Lemma \ref{lemma-product-representable-transformations-property} the morphism $U \times_S U \to F \times F$ is surjective and \'etale (note also that \'etale and surjective occur in the lists of Remarks \ref{remark-list-properties-fpqc-local-base} and \ref{remark-list-properties-stable-composition}). It follows either from Lemma \ref{lemma-base-change-representable-transformations} and the diagram above, or by writing $R \to F$ as $R \to U \to F$ and Lemmas \ref{lemma-morphism-schemes-gives-representable-transformation} and \ref{lemma-composition-representable-transformations} that $R \to F$ is representable as well. Let $T$ be a scheme and let $a : T \to F \times F$ be a morphism. We have to show that $G = T \times_{a, F \times F, \Delta} F$ is representable. By what was said above the morphism (of schemes) $$ T' = (U \times_S U) \times_{F \times F, a} T \longrightarrow T $$ is surjective and \'etale. Hence $\{T' \to T\}$ is an \'etale covering of $T$. Note also that $$ T' \times_T G = T' \times_{U \times_S U, j} R $$ as can be seen contemplating the following cube $$ \xymatrix{ & R \ar[rr] \ar[dd] & & F \ar[dd] \\ T' \times_T G \ar[rr] \ar[dd] \ar[ru] & & G \ar[dd] \ar[ru] & \\ & U \times_S U \ar'[r][rr] & & F \times F \\ T' \ar[rr] \ar[ru] & & T \ar[ru] } $$ Hence we see that the restriction of $G$ to $T'$ is representable by a scheme $X$, and moreover that the morphism $X \to T'$ is a base change of the morphism $j$. Hence $X \to T'$ is separated and locally quasi-finite (see second paragraph of the proof). By Descent, Lemma \ref{descent-lemma-descent-data-sheaves} we obtain a descent datum $(X, \varphi)$ relative to the fppf-covering $\{T' \to T\}$. Since $X \to T$ is separated and locally quasi-finite we see by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} that this descent datum is effective. Hence by Descent, Lemma \ref{descent-lemma-descent-data-sheaves} (2) we conclude that $G$ is representable as desired.