Lemma 89.27.4. 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 89.21.1. 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 89.27.2 we conclude that $d\phi$ is injective (hence bijective by dimension reasons). Thus $\phi : U \to U$ is an isomorphism by Lemma 89.27.3. We can use a similar argument, using the exact sequence

$0 \to \text{Inf}([U/R]) \to TR \to TU \oplus TU$

of Lemma 89.26.2 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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