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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