Carrot is the one who finds the grubby piece of paper first, some time before the lad was promoted to Captain.
He shows it first to Sgt. Colon, who scratches his head and asks Carrot why he's showing it to Colon. Carrot tells him that it is important to foster good relations between colleagues as that is the very Foundation of the Trust that must imbue the Watch as and when Ankh Morpork's finest officers will boldly Keep the Peace should Dangerous Situations arise, and so they must help Constable Nobbs achieve his dreams in whatever way they can, as Nobby would do the same for any of them.
Fred Colon considers the bit about Dangerous Situations and carefully suggests taking the list to Mister Vimes. Carrot does indeed do so, in a manner such that Vimes can't immediately throw his newest recruit out, which is not in the least due to the man's utter sincerity. He can't promise anything immediately, of course, and won't Nobby miss the piece of paper?
Possibly, sir, Carrot says, which is why I took the liberty of copying the list. He salutes smartly and leaves Vimes with the copy. There are many things Vimes has always meant to do, including paperwork, which is still the bane of his life, and the question of Lady Sybil Ramkin, who is firmly convinced he is good enough for her despite all evidence to the contrary. Staring at the piles he has accumulated on his desk, he sighs and puts the list to the side.
Over the years, Watchmen come into the Watch, first as a trickle of Cuddy, Detritus, Angua, then more stream in with Cheery, Reg, Washpot, Wee Mad Arthur until the city is full to the bursting with Sammies and they have travelled all the way to Uberwald. Vimes grumbles, and Carrot remains his right hand man. Colon and Nobbs remain the faithful old troopers of the old guard; occasionally they may seem even fatherly or perhaps more dirty-old-uncles-ish to the new recruits. Be a Man in the City Watch, they proclaim. Sam marries Sybil, and Carrot fills Angua in on age old traditions, and the little known facts of practically everything. Reg Shoes buries himself in the ground every May, Washpot hands out religious pamphlets, Cheery leads the female dwarfs in expressing their feminity as they wish to, Detritus leads the new recruits around Pseudopolis Yard in a groundshaking jog, and in the evenings he cares fondly for his Piecemaker. The City Watch grows and grows.
Barely anyone cares overly much any more than Captain Vimes and his small band of Watchmen did when Nobby steals the sugar cubes out of the mess, nor do they notice when he carefully crosses things off a grubby piece of paper--
Ask Hogfather for armoury outlined in weapons catalogue.
Get a signed bit of paper showing 'Nobby Nobbs is human.'
Travel to the desert (research for production of Ankh Morpork: Before the City Walls for The Peeled Nuts).
Attend a posh ball.
Not getting a promotion.
--save of course, a sharp-eyed Captain and by extension, his Commander. "And that's Nobby done, is it?" Vimes asks, lighting a cigar.
"Yes, sir," Carrot says. "Cheery wants a new set of testtubes."
"They do get broken rather frequently, sir," Carrot points out.
"She's my best Forensic officer; if she wants them, she can have them," Vimes sighs. He stares keenly at Carrot. "I notice that with all these years of keeping Watchmen happy in the little ways, you've never once played fairy godmother for me."
"Did you want me to, sir?" Carrot asks, all sincerity.
"No!" But he does glance around at the paperwork piles, which, though they have never lessened because that would take a miracle, have never quite grown to impossible heights either.