stored procedure to input data in a join table?

There is a many to many between cars and owners for rental
purposes so I added the company_car join table.

company:
companyId
company_type [owner, lessee]

company_car_join:
carID
companyID

car_owner:
carID
companyID
start_date
end_date

So I think I need a SQL query or a procedure to update the join table
from the car_owner table? This is as far as I got on the stored
procedure. If I need a SQL update query I don't know how to write it.

CREATE PROCEDURE update_car_comp_jn()

Begin
for each row in Comp_Car_Owner
set car_comp_join.comp_id = car_owner.comp_id
set car_comp_join.car_id = car_owner.car_id

END

I have one question on structure. Is it right I don't need an auto-
increment key since I can use the concatenated company_id and the
car_id as a unique key? I can drop the column company_id from
car_owner table.

thanks,