By Groupoids, Lemma \ref{groupoids-lemma-restrict-relation} the morphism $j' = (t', s') : R' \to U' \times_S U'$ defines an equivalence relation. Since $g$ is flat and locally of finite presentation we see that $g$ is universally open as well (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}). For the same reason $s, t$ are universally open as well. Let $W^1 = g(U') \subset U$, and let $W = t(s^{-1}(W^1))$. Then $W^1$ and $W$ are open in $U$. Moreover, as $j$ is an equivalence relation we have $t(s^{-1}(W)) = W$ (see Groupoids, Lemma \ref{groupoids-lemma-constructing-invariant-opens} for example). \medskip\noindent By Groupoids, Lemma \ref{groupoids-lemma-quotient-pre-equivalence-relation-restrict} the map of sheaves $F' = U'/R' \to F = U/R$ is injective. Let $a : T \to F$ be a morphism from a scheme into $U/R$. We have to show that $T \times_F F'$ is representable by an open subscheme of $T$. \medskip\noindent The morphism $a$ is given by the following data: an fppf covering $\{\varphi_j : T_j \to T\}_{j \in J}$ of $T$ and morphisms $a_j : T_j \to U$ such that the maps $$ a_j \times a_{j'} : T_j \times_T T_{j'} \longrightarrow U \times_S U $$ factor through $j : R \to U \times_S U$ via some (unique) maps $r_{jj'} : T_j \times_T T_{j'} \to R$. The system $(a_j)$ corresponds to $a$ in the sense that the diagrams $$ \xymatrix{ T_j \ar[r]_{a_j} \ar[d] & U \ar[d] \\ T \ar[r]^a & F } $$ commute. \medskip\noindent Consider the open subsets $W_j = a_j^{-1}(W) \subset T_j$. Since $t(s^{-1}(W)) = W$ we see that $$ W_j \times_T T_{j'} = r_{jj'}^{-1}(t^{-1}(W)) = r_{jj'}^{-1}(s^{-1}(W)) = T_j \times_T W_{j'}. $$ By Descent, Lemma \ref{descent-lemma-open-fpqc-covering} this means there exists an open $W_T \subset T$ such that $\varphi_j^{-1}(W_T) = W_j$ for all $j \in J$. We claim that $W_T \to T$ represents $T \times_F F' \to T$. \medskip\noindent First, let us show that $W_T \to T \to F$ is an element of $F'(W_T)$. Since $\{W_j \to W_T\}_{j \in J}$ is an fppf covering of $W_T$, it is enough to show that each $W_j \to U \to F$ is an element of $F'(W_j)$ (as $F'$ is a sheaf for the fppf topology). Consider the commutative diagram $$ \xymatrix{ W'_j \ar[rr] \ar[dd] \ar[rd] & & U' \ar[d]^g \\ & s^{-1}(W^1) \ar[r]_s \ar[d]^t & W^1 \ar[d] \\ W_j \ar[r]^{a_j|_{W_j}} & W \ar[r] & F } $$ where $W'_j = W_j \times_W s^{-1}(W^1) \times_{W^1} U'$. Since $t$ and $g$ are surjective, flat and locally of finite presentation, so is $W'_j \to W_j$. Hence the restriction of the element $W_j \to U \to F$ to $W'_j$ is an element of $F'$ as desired. \medskip\noindent Suppose that $f : T' \to T$ is a morphism of schemes such that $a|_{T'} \in F'(T')$. We have to show that $f$ factors through the open $W_T$. Since $\{T' \times_T T_j \to T'\}$ is an fppf covering of $T'$ it is enough to show each $T' \times_T T_j \to T$ factors through $W_T$. Hence we may assume $f$ factors as $\varphi_j \circ f_j : T' \to T_j \to T$ for some $j$. In this case the condition $a|_{T'} \in F'(T')$ means that there exists some fppf covering $\{\psi_i : T'_i \to T'\}_{i \in I}$ and some morphisms $b_i : T'_i \to U'$ such that $$ \xymatrix{ T'_i \ar[r]_{b_i} \ar[d]_{f_j \circ \psi_i} & U' \ar[r]_g & U \ar[d] \\ T_j \ar[r]^{a_j} & U \ar[r] & F } $$ is commutative. This commutativity means that there exists a morphism $r'_i : T'_i \to R$ such that $t \circ r'_i = a_j \circ f_j \circ \psi_i$, and $s \circ r'_i = g \circ b_i$. This implies that $\Im(f_j \circ \psi_i) \subset W_j$ and we win.