Over heating light

Question

I have an 2004 atv Arctic cat 400 auto. I ran in a hot day and my o-h light was on but the fan did not start before. He started very late and he got off with the o-h light still on. I made the road remaining (~10 miles) at high speed to vent the radiator always with the o-h light on. Was it dangerous for my motor/transmission and where is the problem?