Lemma 84.29.3. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Let $X$ be the simplicial scheme over $S$ constructed in Lemma 84.29.1. Let $(R/U)_\bullet $ be the simplicial scheme associated to $s : R \to U$, see Definition 84.27.3. There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$ of simplicial schemes with low degree morphisms given by

**Proof.**
For arbitrary $n$ we define $(R/U)_\bullet \to X_ n$ by the rule

Compatibility with degeneracy maps is clear from the description of the degeneracies in Lemma 84.29.1. We omit the verification that the maps respect the morphisms $s^ n_ j$. Groupoids, Lemma 39.13.5 (with the roles of $s$ and $t$ reversed) shows that the two right squares are cartesian. In exactly the same manner one shows all the other squares are cartesian too. Hence the morphism is cartesian. $\square$

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