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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