I think 212 is getting close - as he says work is done on the air.
However if you take my previous example of the piston and cylinder, although you do work on the air to move the piston to compress, you could then connect the piston to some machine and have the compressed air do work on the machine as it expands, getting all your energy back. But in the case of the heater, you have to do more work than the minimum required to compress the air in order to overcome the inefficiencies / friction, and it is that bit of the work only, that results in hot air in the cockpit.
Another example - car on a hilly road. On the way up hill, the engine has to work hard to increase the potential energy of the car. On the way back down the other side, the potential energy is released and means the engine has to add no power to keep the car moving. But overall, the engine still has to produce power to overcome friction (tyres, bearings, air etc). That additional energy ultimately goes to heating up those tyres, bearings, the air etc.
So 212, the compression could take place "for free" if the process was 100%efficient and if the energy released in expansion was captured. But of course its not, so the air ends up hotter.
HC