In the second part of the proof for 10.160.8 (the case of positive characteristic), the terminology is slightly confusing. I suggest rewording the condition required of to the commutative square , where are the projections and , and is a chosen isomorphism. Moreover, the map to lift using formal smoothness is the composition .
It could also be helpful for readers to mention that this proof shows that the map from Cohen rings (up to isomorphism) to fields (up to isomorphism) is injective.

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