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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