Based on the NEC, under the right situation, it is perfectly legal to place a 10AWG conductor in a 60AMP breaker, under the right conditions I could place a 10AWG conductor in a 250AMP breaker, but those are for motor circuits, you said you have an AC unit. There is an article in the NEC which deals with AC units Article 440, you do not size conductors/overloads/disconnects for AC units as you do for motors.
Generally (meaning I can’t see your installation, there might be exceptions), the rating of the conductors and equipment is based on the rated-load current on the nameplate (there are exceptions). The short-circuit and ground-fault protection shall not exceed 175% of the nameplate, unless this is not high enough to carry the starting current of the motor, in which case you can increase to 225%. The conductors shall have an ampacity of not less than 125%.
So is it possible to place a 10AWG conductor in a 60AMP breaker used on an AC unit? Yes.
Is it ok to do so in your installation? I don’t know, I can’t see your installation.