Situation 40.11.1. Let $S$ be a scheme. Let $U = \mathop{\mathrm{Spec}}(k)$ be a scheme over $S$ with $k$ a field. Let $(U, R_1, s_1, t_1, c_1)$, $(U, R_2, s_2, t_2, c_2)$ be groupoid schemes over $S$ with identical first component. Let $a : R_1 \to R_2$ be a morphism such that $(\text{id}_ U, a)$ defines a morphism of groupoid schemes over $S$, see Groupoids, Definition 39.13.1. In particular, the following diagrams commute

\[ \vcenter { \xymatrix{ R_1 \ar[rrd]^{t_1} \ar[rdd]_{s_1} \ar[rd]_ a \\ & R_2 \ar[d]^{t_2} \ar[r]_{s_2} & U \\ & U } } \quad \quad \vcenter { \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r]_-{c_1} \ar[d]_{a \times a} & R_1 \ar[d]^ a \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r]^-{c_2} & R_2 } } \]

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