Keep in mind that setting the dial to 80 doesn't make it blow 80 degree air, and setting it to 60 doesn't make it blow 60 degree air. That's not how refrigeration works. When the compressor is on, the interior air conditioner coils are cold. Not 60 degrees or 80 degrees. Just cold. I'm not sure how cold. A wild guess is 40 degrees. The air won't be that cold, but the coils will be.
The air conditioner blows cold air until the interior thermostat tells it the interior air is cool enough to turn off the compressor. If you have the fan set to a manual speed, at that point, the system will start blowing outside air at outside temperatures (unless you have it set to recycle air, in which case, it is just blowing around interior air). It will do this until the inside thermometer tells the system the interior is getting too warm, at which point, the compressor turns on again and the coils get cold.
If you don't have the fan on manual, the system also changes the fan setting, so if it is a lot hotter inside the car than the desired temperature, the fan is on high, and the closer you get to the temperature you want, the slower the fan is blowing. This is just a way to mask the "on/off" nature of refrigeration as we know it.
Keep in mind that turning off the compressor does not immediately stop refrigeration. You'll still have a few seconds of refrigeration as the higher pressure between the compressor and the coil bleeds down through the rest of the system.
Once the refrigeration gasses have stopped chilling the coils, remember that the coils are made of metal and they will still be cold. You'll get a cooling effect until their thermal mass is brought up to the temperature of the air blowing through them.
This is quite enough to explain the chunk of a minute that you still get cold air blowing after you've turned the AC off. Be guilt free.