Lemma 89.4.2. Let $f: R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda $. The following are equivalent

$f$ is surjective,

the map $\mathfrak m_ R/\mathfrak m_ R^2 \to \mathfrak m_ S/\mathfrak m_ S^2$ is surjective, and

the map $\mathfrak m_ R/(\mathfrak m_\Lambda R + \mathfrak m_ R^2) \to \mathfrak m_ S/(\mathfrak m_\Lambda S + \mathfrak m_ S^2)$ is surjective.

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