It seems UA has just a size specification and no weight limit. Remarkable. You could print that out.
Also maybe print the Air NZ Gold carry-on allowance:
Allowance
•Economy fares get one carry-on bag plus one small item, like a handbag or thin laptop.
•Premium Economy, Business Premier, connecting Business Premier, and Airpoints™ Gold, Elite and Star Alliance Gold members get two bags plus one small item.
Weight
•Economy carry-on can weigh up to 7kg (15lb).
•Premium Economy, Business Premier, connecting Business Premier, and Airpoints™ Gold, Elite and Star Alliance Gold members' carry-on can weigh up to 14kg (30lb) total, with one item weighing up to 10kg (22lb).