Lemma 89.27.7. Let $\mathcal{F}$ be category cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume there exist presentations of $\mathcal{F}$ by minimal smooth prorepresentable groupoids in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$. Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic.

Proof. Follows from Lemma 89.27.5 and the observation that a morphism $[U/R] \to [U'/R']$ is the same thing as a morphism of groupoids in functors (by our explicit construction of $[U/R]$ in Definition 89.21.9). $\square$

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