The Stacks project

Lemma 89.8.11. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$. For any formal object $\eta $ of $\widehat{\mathcal{F}}$, there exists a morphism $\xi \to \eta $.

Proof. By assumption the morphism $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ is smooth. Then $\iota (\xi ) : \underline{R} \to \widehat{\mathcal{F}}$ is the completion of $\underline{\xi }$, see Remark 89.7.12. By Lemma 89.8.8 there exists an object $f$ of $\underline{R}$ such that $\iota (\xi )(f) = \eta $. Then $f$ is a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda $. And $\iota (\xi )(f) = \eta $ means that $f_*\xi \cong \eta $ which means exactly that there is a morphism $\xi \to \eta $ lying over $f$. $\square$

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