Yes, cold air is provided by the AC system and warm/hot air by the vehicle’s engine cooling system so even with no AC working you will still get warm air when selected.
You could also run a few DIY checks, see if the engine revs drop slightly when you switch on the AC and check if the fan(s) cut in and out with the AC switched on.
An auto AC specialist with the correct diagnostic equipment would be your best bet, someone like Diagnose Dan!