Conditions (1), (2), (3), and (4) and Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} imply the affine scheme $\overline{U}$ representing $U/P$ exists, the morphism $U \to \overline{U}$ is finite locally free, and $P = U \times_{\overline{U}} U$. The identification $P = U \times_{\overline{U}} U$ is such that $t|_P = \text{pr}_0$ and $s|_P = \text{pr}_1$, and such that composition is equal to $\text{pr}_{02} : U \times_{\overline{U}} U \times_{\overline{U}} U \to U \times_{\overline{U}} U$. A product of finite locally free morphisms is finite locally free (see Spaces, Lemma \ref{spaces-lemma-product-representable-transformations-property} and Morphisms, Lemmas \ref{morphisms-lemma-base-change-finite-locally-free} and \ref{morphisms-lemma-composition-finite-locally-free}). To get $\overline{R}$ we are going to descend the scheme $R$ via the finite locally free morphism $U \times_S U \to \overline{U} \times_S \overline{U}$. Namely, note that $$ (U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) = P \times_S P $$ by the above. Thus giving a descent datum (see Descent, Definition \ref{descent-definition-descent-datum}) for $R / U \times_S U / \overline{U} \times_S \overline{U}$ consists of an isomorphism $$ \varphi : R \times_{(U \times_S U), t \times t} (P \times_S P) \longrightarrow (P \times_S P) \times_{s \times s, (U \times_S U)} R $$ over $P \times_S P$ satisfying a cocycle condition. We define $\varphi$ on $T$-valued points by the rule $$ \varphi : (r, (p, p')) \longmapsto ((p, p'), p^{-1} \circ r \circ p') $$ where the composition is taken in the groupoid category $(U(T), R(T), s, t, c)$. This makes sense because for $(r, (p, p'))$ to be a $T$-valued point of the source of $\varphi$ it needs to be the case that $t(r) = t(p)$ and $s(r) = t(p')$. Note that this map is an isomorphism with inverse given by $((p, p'), r') \mapsto (p \circ r' \circ (p')^{-1}, (p, p'))$. To check the cocycle condition we have to verify that $\varphi_{02} = \varphi_{12} \circ \varphi_{01}$ as maps over $$ (U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) = (P \times_S P) \times_{s \times s, (U \times_S U), t \times t} (P \times_S P) $$ By explicit calculation we see that $$ \begin{matrix} \varphi_{02} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), (p_1 \circ p_2)^{-1} \circ r \circ (p_1' \circ p_2')) \\ \varphi_{01} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), p_1^{-1} \circ r \circ p_1', (p_2, p_2')) \\ \varphi_{12} & ((p_1, p_1'), r, (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), p_2^{-1} \circ r \circ p_2') \end{matrix} $$ (with obvious notation) which implies what we want. As $j$ is separated and locally quasi-finite by (5) we may apply More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} to get a scheme $\overline{R} \to \overline{U} \times_S \overline{U}$ and an isomorphism $$ R \to \overline{R} \times_{(\overline{U} \times_S \overline{U})} (U \times_S U) $$ which identifies the descent datum $\varphi$ with the canonical descent datum on $\overline{R} \times_{(\overline{U} \times_S \overline{U})} (U \times_S U)$, see Descent, Definition \ref{descent-definition-effective}. \medskip\noindent Since $U \times_S U \to \overline{U} \times_S \overline{U}$ is finite locally free we conclude that $R \to \overline{R}$ is finite locally free as a base change. Hence $R \to \overline{R}$ is surjective as a map of sheaves on $(\Sch/S)_{fppf}$. Our choice of $\varphi$ implies that given $T$-valued points $r, r' \in R(T)$ these have the same image in $\overline{R}$ if and only if $p^{-1} \circ r \circ p'$ for some $p, p' \in P(T)$. Thus $\overline{R}$ represents the sheaf $$ T \longmapsto  \overline{R(T)} = P(T)\backslash R(T)/P(T) $$ with notation as in the discussion preceding the lemma. Hence we can define the groupoid structure on $(\overline{U} = U/P, \overline{R} = P\backslash R/P)$ exactly as in the discussion of the ``plain'' groupoid case. It follows from this that $(U, R, s, t, c)$ is the pullback of this groupoid structure via the morphism $U \to \overline{U}$. This concludes the proof.