Lemma 90.27.4. Let (U, R, s, t, c) be a minimal smooth prorepresentable groupoid in functors on \mathcal{C}_\Lambda . If \varphi : [U/R] \to [U/R] is an equivalence of categories cofibered in groupoids, then \varphi is an isomorphism.
Proof. A morphism \varphi : [U/R] \to [U/R] is the same thing as a morphism \varphi : (U, R, s, t, c) \to (U, R, s, t, c) of groupoids in functors over \mathcal{C}_\Lambda as defined in Definition 90.21.1. Denote \phi : U \to U and \psi : R \to R the corresponding morphisms. Because the diagram
\xymatrix{ & \text{Der}_\Lambda (k, k) \ar[dr]_\gamma \ar[dl]^\gamma \\ TU \ar[rr]_{d\phi } \ar[d] & & TU \ar[d] \\ T[U/R] \ar[rr]^{d\varphi } & & T[U/R] }
is commutative, since d\varphi is bijective, and since we have the characterization of minimality in Lemma 90.27.2 we conclude that d\phi is injective (hence bijective by dimension reasons). Thus \phi : U \to U is an isomorphism by Lemma 90.27.3. We can use a similar argument, using the exact sequence
0 \to \text{Inf}([U/R]) \to TR \to TU \oplus TU
of Lemma 90.26.2 to prove that \psi : R \to R is an isomorphism. But is also a consequence of the fact that R = U \times _{[U/R]} U and that \varphi and \phi are isomorphisms. \square
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