We only hear about Kuzyomin once at the very beginning of the book, but he basically provides us with Rules For Living in the Camp. He warned Shukhov back in the day about signs of doom for a zek: if a zek licks bowls, trusts in sickbay, or acts as a snitch, then he won't last long. Kuzyomin helps us to grasp how long Shukhov has been in the gulag system and how much he's learned since his early days.
Kuzyomin's warning signs are also running themes throughout the book. We hear about Fetyukov getting beaten for licking bowls at the end. We hear about Shukhov's pointless trip to sickbay and his eventual decision to avoid it for his own safety. And we hear about how Pantleyev, the snitch, is hated by his whole gang, which doesn't bode well for his safety. Overall, all of Kuzyomin's warning signs have to do with having false hope or trusting the wrong people and the wrong things to survive.