co-author vs writing

co-author is not a noun while writing is a noun.

Word	Noun	Adjective	Verb	Adverb
co-author	No	No	Yes	No
writing	Yes	No	No	No

co-author (noun)	writing (noun)
	the activity of putting something in written form
	the act of creating written works
	the work of a writer; anything expressed in letters of the alphabet (especially when considered from the point of view of style and effect)
	letters or symbols that are written or imprinted on a surface to represent the sounds or words of a language
	(usually plural) the collected work of an author

co-author (verb)	writing (verb)
be a co-author on (a book, a paper)