The Stacks project

Remark 88.22.3. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda $. Then there is a canonical isomorphism $(U, R, s, t, c)^{\wedge }|_{\mathcal{C}_\Lambda } \cong (U, R, s, t, c)$, see Remark 88.7.7. On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on $\widehat{\mathcal{C}}_\Lambda $ such that $U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}$ both commute with limits, e.g. if $U, R$ are representable. Then there is a canonical isomorphism $((U, R, s, t, c)|_{\mathcal{C}_\Lambda })^{\wedge } \cong (U, R, s, t, c)$. This follows from Remark 88.7.11.


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