Remark 90.7.11. Let G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets} be a functor that commutes with limits. Then the map G \to \widehat{G|_{\mathcal{C}_\Lambda }} described in Remark 90.7.9 is an isomorphism. Indeed, if S is an object of \widehat{\mathcal{C}}_\Lambda , then we have canonical bijections
\widehat{G|_{\mathcal{C}_\Lambda }}(S) = \mathop{\mathrm{lim}}\nolimits _ n G(S/\mathfrak {m}_ S^ n) = G(\mathop{\mathrm{lim}}\nolimits _ n S/\mathfrak {m}_ S^ n) = G(S).
In particular, if R is an object of \widehat{\mathcal{C}}_\Lambda then \underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda }} because the representable functor \underline{R} commutes with limits by definition of limits.
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