Lemma 39.13.5. Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. The diagram

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

Lemma 39.13.5. Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. The diagram

39.13.5.1

\begin{equation} \label{groupoids-equation-pull} \xymatrix{ R \times _{t, U, t} R \ar@<1ex>[r]^-{\text{pr}_1} \ar@<-1ex>[r]_-{\text{pr}_0} \ar[d]_{(\text{pr}_0, 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.

**Proof.**
The commutativity of the diagram follows from the axioms of a groupoid. Note that, in terms of groupoids, the top left vertical arrow assigns to a pair of morphisms $(\alpha , \beta )$ with the same target, the pair of morphisms $(\alpha , \alpha ^{-1} \circ \beta )$. In any groupoid this defines a bijection between $\text{Arrows} \times _{t, \text{Ob}, t} \text{Arrows}$ and $\text{Arrows} \times _{s, \text{Ob}, t} \text{Arrows}$. Hence the second assertion of the lemma. The last assertion follows from Lemma 39.13.4.
$\square$

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