The Stacks project

Lemma 79.9.7. In Situation 79.9.2. If $r : \mathop{\mathrm{Spec}}(k) \to R$ is a morphism such that $s \circ r, t \circ r$ are automorphisms of $\mathop{\mathrm{Spec}}(k)$, then the map

\[ R \longrightarrow R, \quad x \longmapsto c(r, x) \]

is an automorphism $R \to R$ which maps $e$ to $r$.

Proof. Proof is identical to the proof of More on Groupoids, Lemma 40.10.6. $\square$

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