Lemma 89.27.5. Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth prorepresentable groupoids 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. Let $\psi : [U'/R'] \to [U/R]$ be a quasi-inverse to $\varphi$. Then $\psi \circ \varphi$ and $\varphi \circ \psi$ are isomorphisms by Lemma 89.27.4, hence $\varphi$ and $\psi$ are isomorphisms. $\square$

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