Let us denote $f : U \to S$ the structure morphism of $U$. By assumption (6) we can write $s^{-1}(\{u\}) = \{r_1, \ldots, r_n\}$. Since this set is finite, we see that $s$ is quasi-finite at each of these finitely many inverse images, see Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}. Hence we see that $f \circ s : R \to S$ is quasi-finite at each $r_i$ (Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-finite}). Hence $r_i$ is isolated in the fibre $R_p$, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}. Write $t(\{r_1, \ldots, r_n\}) = \{u_1, \ldots, u_m\}$. Note that it may happen that $m < n$ and note that $u \in \{u_1, \ldots, u_m\}$. Since $t$ is flat and locally of finite presentation, the morphism of fibres $t_p : R_p \to U_p$ is flat and locally of finite presentation (Morphisms, Lemmas \ref{morphisms-lemma-base-change-flat} and \ref{morphisms-lemma-base-change-finite-presentation}), hence open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}). The fact that each $r_i$ is isolated in $R_p$ implies that each $u_j = t(r_i)$ is isolated in $U_p$. Using Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize} again, we see that $f$ is quasi-finite at $u_1, \ldots, u_m$. \medskip\noindent Denote $F_u = s^{-1}(u)$ and $F_{u_j} = s^{-1}(u_j)$ the scheme theoretic fibres. Note that $F_u$ is finite over $\kappa(u)$ as it is locally of finite type over $\kappa(u)$ with finitely many points (for example it follows from the much more general Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded}). By Lemma \ref{lemma-two-fibres} we see that $F_u$ and $F_{u_j}$ become isomorphic over a common field extension of $\kappa(u)$ and $\kappa(u_j)$. Hence we see that $F_{u_j}$ is finite over $\kappa(u_j)$. In particular we see $s^{-1}(\{u_j\})$ is a finite set for each $j = 1, \ldots, m$. Thus we see that assumptions (2) and (6) hold for each $u_j$ also (above we saw that $U \to S$ is quasi-finite at $u_j$). Hence the argument of the first paragraph applies to each $u_j$ and we see that $R \to U$ is quasi-finite at each of the points of $$ \{r_1, \ldots, r_N\} = s^{-1}(\{u_1, \ldots, u_m\}) $$ Note that $t(\{r_1, \ldots, r_N\}) = \{u_1, \ldots, u_m\}$ and $t^{-1}(\{u_1, \ldots, u_m\}) = \{r_1, \ldots, r_N\}$ since $R$ is a groupoid\footnote{Explanation in groupoid language: The original set $\{r_1, \ldots, r_n\}$ was the set of arrows with source $u$. The set $\{u_1, \ldots, u_m\}$ was the set of objects isomorphic to $u$. And $\{r_1, \ldots, r_N\}$ is the set of all arrows between all the objects equivalent to $u$.}. Moreover, we have $\text{pr}_0(c^{-1}(\{r_1, \ldots, r_N\})) = \{r_1, \ldots, r_N\}$ and $\text{pr}_1(c^{-1}(\{r_1, \ldots, r_N\})) = \{r_1, \ldots, r_N\}$. Similarly we get $e(\{u_1, \ldots, u_m\}) \subset \{r_1, \ldots, r_N\}$ and $i(\{r_1, \ldots, r_N\}) = \{r_1, \ldots, r_N\}$. \medskip\noindent We may apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical} to the pairs $(U \to S, \{u_1, \ldots, u_m\})$ and $(R \to S, \{r_1, \ldots, r_N\})$ to get an \'etale neighbourhood $(S', p') \to (S, p)$ which induces an identification $\kappa(p) = \kappa(p')$ such that $S' \times_S U$ and $S' \times_S R$ decompose as $$ S' \times_S U = U' \amalg W, \quad S' \times_S R = R' \amalg W' $$ with $U' \to S'$ finite and $(U')_{p'}$ mapping bijectively to $\{u_1, \ldots, u_m\}$, and $R' \to S'$ finite and $(R')_{p'}$ mapping bijectively to $\{r_1, \ldots, r_N\}$. Moreover, no point of $W_{p'}$ (resp.\ $(W')_{p'}$) maps to any of the points $u_j$ (resp.\ $r_i$). At this point (a), (b), (c), and (d) of the lemma are satisfied. Moreover, the inclusions of (e) and (f) hold on fibres over $p'$, i.e., $s'((R')_{p'}) \subset (U')_{p'}$, $t'((R')_{p'}) \subset (U')_{p'}$, and $c'((R' \times_{s', U', t'} R')_{p'}) \subset (R')_{p'}$. \medskip\noindent We claim that we can replace $S'$ by a Zariski open neighbourhood of $p'$ so that the inclusions of (e) and (f) hold. For example, consider the set $E = (s'|_{R'})^{-1}(W)$. This is open and closed in $R'$ and does not contain any points of $R'$ lying over $p'$. Since $R' \to S'$ is closed, after replacing $S'$ by $S' \setminus (R' \to S')(E)$ we reach a situation where $E$ is empty. In other words $s'$ maps $R'$ into $U'$. Note that this property is preserved under further shrinking $S'$. Similarly, we can arrange it so that $t'$ maps $R'$ into $U'$. At this point (e) holds. In the same manner, consider the set $E = (c'|_{R' \times_{s', U', t'} R'})^{-1}(W')$. It is open and closed in the scheme $R' \times_{s', U', t'} R'$ which is finite over $S'$, and does not contain any points lying over $p'$. Hence after replacing $S'$ by $S' \setminus (R' \times_{s', U', t'} R' \to S')(E)$ we reach a situation where $E$ is empty. In other words we obtain the inclusion in (f). We may repeat the argument also with the identity $e' : S' \times_S U \to S' \times_S R$ and the inverse $i' : S' \times_S R \to S' \times_S R$ so that we may assume (after shrinking $S'$ some more) that $(e'|_{U'})^{-1}(W') = \emptyset$ and $(i'|_{R'})^{-1}(W') = \emptyset$. \medskip\noindent At this point we see that we may consider the structure $$ (U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times_{t', U', s'} R'}, e'|_{U'}, i'|_{R'}). $$ The axioms of a groupoid scheme over $S'$ hold because they hold for the groupoid scheme $(S' \times_S U, S' \times_S R, s', t', c', e', i')$.