Proof.
Conditions (1), (2), (3), and (4) and Groupoids, Proposition 39.23.9 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 65.5.7 and Morphisms, Lemmas 29.48.4 and 29.48.3). 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 35.34.1) 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 37.57.1 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 35.34.10.
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 (\mathit{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.
\square
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