Should Nazi be capitalized in the phrase grammar nazi/Nazi?
While I can't think of any other examples right now, I would like to extend the question to ask if the words which are historically nouns should be capitalized in similar phrases.
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I would recommend lowercase, if you are not actually referring to the Nazi party, its members, or principles.
Either one works.
Whether you capitalize Nazi or leave it generically uncapitalized depends on how much you want to emphasize it, and how much respect you want to appear to show for the NSDAP.
Oh, and this is not about grammatical correctness. Grammar has nothing to do with literacy issues like spelling, punctuation, or capitalization. Grammar has to do with spoken language, and these issues don't arise there.
As "Nazi" is an acronym and a proper noun, I think it should be capitalized. Whether or not you are referring to the literal Nazi party, that is still where the name comes from. Like, "French" refers to the nation of France, but even when you are using the word in a way that is not literally referring to that nation -- "French bread", "a French-sounding name", etc -- you still capitalize it.