sponsor vs cosponsor

sponsor is a noun but cosponsor is not a noun.

Word	Noun	Adjective	Verb	Adverb
sponsor	Yes	No	Yes	No
cosponsor	No	No	Yes	No

As verbs, cosponsor is a hyponym of sponsor; that is, cosponsor is a word with a more specific, narrower meaning than sponsor:

sponsor (noun)	cosponsor (noun)
someone who supports or champions something
an advocate who presents a person (as for an award or a degree or an introduction etc.)

sponsor (verb)	cosponsor (verb)
assume sponsorship of	sponsor together with another sponsor
assume responsibility for or leadership of
do one's shopping at; do business with; be a customer or client of