The pilot flys for the Company that employs him. In your example, that would be Chautauqua.
The airline, again Chautauqua, assigns the pilot to particular flights based on his "bid" for a "line of time (flying) which is awarded on the basis of seniority at his airline.
Cautauqua (not the pilot) contracts with several different companies, but the pilot is employed by one company only; in this case Chautauqua.
Good question.