Fan and heat problem

Hello, when I start my 2012 model peugeot308 1.6 vti access gasoline car, the fan turns on after 10 seconds even when the engine is cold and it overheats. They said that I had the Thermostat checked in the private service, they said it was solid, they said the brain of the car might be malfunctioning. I took the car to the authorized service, they said that the water pump in the service was sweating and the thermostat would change. Do you think it can be from the water pump, will there be software in the brain of the car or will the brain change? If I made the water pump, will the malfunction go away? Thanks

is there any water reduction?

no no water reduction

if the water does not decrease, then it is a thermostat. what do you think it has to do with the brain? the best thing to do in the service

Yes I did. I put the car in service. I hope it won't bother you there