authorship vs co-author

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

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

authorship (noun)	co-author (noun)
the act of creating written works
the act of initiating a new idea or theory or writing

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