Remark 97.21.4 (Automorphisms). Assumptions and notation as in Lemma 97.21.2. Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition map

$\text{Inf}(x'/x) \times \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'') \times \text{Inf}(x''/x) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'').$

Since $\textit{Lift}(x, A')$ is a groupoid, if $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines a simply transitive left action of $\text{Inf}(x'/x)$ on $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ and a simply transitive right action by $\text{Inf}(x''/x)$. Now the lemma says that $\text{Inf}(x'/x) = \text{Inf}_ x(I) = \text{Inf}(x''/x)$. We claim that the two actions described above agree via these identifications. Namely, either $x' \not\cong x''$ in which the claim is clear, or $x' \cong x''$ and in that case we may assume that $x'' = x'$ in which case the result follows from the fact that $\text{Inf}(x'/x)$ is commutative. In particular, we obtain a well defined action

$\text{Inf}_ x(I) \times \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'') \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$

which is simply transitive as soon as $\mathop{\mathrm{Mor}}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty.

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