You'll have to keep an eye on what comes out of the flue. Some smoke sometimes is normal (when the air adjusts b/c temperatures change, the cat is not instantaneous in being able to consume all additional smoke, as it has to heat up a bit to do so, so there'll be some smoke intermittently), but when you constantly see smoke and when you open the bypass (without changing the air settings) what comes out of the chimney does not change much, then you know your cat is completely done.
Somewhat done means observations somewhat like the above
When you decrease the air (bypass already closed, stove in steady state), the cat temperature generally goes *up* a bit in 15-30 minutes - because there's more smoke to combust . If that does not happen, that's an indication (especially if you see smoke coming out the chimney persistently).
You'll have to get the hang of how the cat behaves. Initially it'll be young and wild ("too hot"), but it'll mature and settle down after - something between 10 burns and a cord... Don't know what that depends on, possibly type of wood (and draft, and thermostat setting, and etc. etc.). Then you're cruising and get used to how it behaves.
At some point it should start changing (for the worse). That is a slow onset. So it's easy to miss it, I presume.
I think I'm (nearly?) there now. We'll see this season.
Others here have gone through many more cats.
But observations do differ quite a lot burning sappy wood (e.g. pine), like
@Highbeam or red oak. I've burned quite some pitch pine (very sappy, used to seal wooden whaling ships in the 1800s here), and a lot of oak. It's different.
So each observation/experience you hear about will be useful, but may not be what you're seeing. Your system and fuel, and running demands will be different than others.
Just pay attention when you burn - as much as possible so you learn, get a feel for how the thing behaves.