## 40.3 Useful diagrams

We briefly restate the results of Groupoids, Lemmas 39.13.4 and 39.13.5 for easy reference in this chapter. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. In the commutative diagram

40.3.0.1
\begin{equation} \label{more-groupoids-equation-diagram} \vcenter { \xymatrix{ & U & \\ R \ar[d]_ s \ar[ru]^ t & R \times _{s, U, t} R \ar[l]^-{\text{pr}_0} \ar[d]^{\text{pr}_1} \ar[r]_-c & R \ar[d]^ s \ar[lu]_ t \\ U & R \ar[l]_ t \ar[r]^ s & U } } \end{equation}

the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.

The diagram

40.3.0.2
\begin{equation} \label{more-groupoids-equation-pull} \vcenter { \xymatrix{ R \times _{t, U, t} R \ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_0 \times c \circ (i, 1)} & R \ar[r]^ t \ar[d]^{\text{id}_ R} & U \ar[d]^{\text{id}_ U} \\ R \times _{s, U, t} R \ar@<1ex>[r]^-c \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{\text{pr}_1} & R \ar[r]^ t \ar[d]^ s & U \\ R \ar@<1ex>[r]^ s \ar@<-1ex>[r]_ t & U } } \end{equation}

is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).