He's 2. Two year olds can be little boggers sometimes, excuse my language. So can 3 year olds.
Cot climbing is more of a behavioural thing than any indication he should change from cot to bed.
There will almost definitely be other behaviour issues you face whereby he's doing something that is unwelcome behaviour - either dangerous or just ill mannered. Some parents face hitting, biting, refusing to hold hands by the road, not sharing, refusing to eat dinner. And a million other unwelcome behaviours that develop in the Terrible Twos and Threenagers do. Cot climbing is no different.
Lots (loads of people I know in real life and also people who wrote threads here) take cot climbing as a sign it's time to move to a bed. To be perfectly honest I think that's just lazy parenting (I know that makes me sound judgmental). It's avoiding dealing with the actual behaviour issue by side-stepping it. It often just creates more of the same when baby then refuses to stay in bed.
So, just deal with it however you are going to deal with other unwelcome behaviours. It rather depends on your parenting style. Are you a negotiator, a friendly briber, an authoritarian, or any of the other ways you could be.
I get my teacher head on. Im quite strict and zero tolerance with things like that. So I'd get cross. Make sure toddler knows that I am not happy with this behaviour, that it is not OK. At the same time going overboard with praise for the behaviour I do want.
At 2 you should also be able to talk to and explain your expectations. Have a simple statement you say over and over again. Something like "you must lie down in the cot". It's good to use poditive statements what you do want, rather than making statements about what you don't want. So I would not use the word climbing.
I would probably start hanging around the doorway at bedtime. Sturn face on. Keep an eye and any attempt to sit up, let alone climb, straight in with a firm "NO! You must lie down to sleep". Stand by the cot if needed. Zero tolerance on anything other than lying down.
Then in the morning, loads of praise for staying lying down at sleep time. Lots of talking about what a good, good boy he is for lying down at sleep time and that you're really proud of the big boy he's growing up to be, lying down his cot like such a good boy.
Lots of smiles and positivity to counter the sturn, cross face when there is any unwelcome behaviour. Ton a very basic level, your son will want to make you happy. So be over the top happy with the behaviour you want. Then be cartoonish unhappy in your face with unwelcome behaviour. Toddlers cotton on to non-verbal communications like this easily, I think.