writing vs co-author

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

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

writing (noun)	co-author (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

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