OK, I'm going to kind of describe a 'typical' air system and the pieces that make it up. We'll see if maybe they left something out that's important to what you're trying to do.
Generally your main pieces are:
Air Tank
Air Compressor
Relay - for powering the compressor
Solenoid - this is the electric valve that controls airflow from the
tank to the horns.
Pressure Switch - This controls when the compressor cuts on and off in relation to tank pressure.
I'm not familiar with the specific items included in the Hadley kit, but most that I have seen have been similar. Generally when I install something like this I use the relay to isolate high current components (the compressor in this case) from anything else that may not be up to the current draw. Some kits come with a relay and some don't, so you may have to get one yourself if you want it.
Anyhow, if I install a system like this I'll use a relay and the wiring will be from some ignition-hot circuit to the pressure switch and from the pressure switch to the coil in the relay. The "load" side of the relay is wired to the compressor and the battery direct through a fuse. The reasons that I do it this way are:
The compressor only runs when the key is in the ignition. No dead batteries if the air tank leaks.
The high-current draw of the compressor only goes through the relay, not the pressure switch (This becomes more important as the current draw of the compressor and the distance from the compressor to the pressure switch increase).
The actual "blowing" of the horn happens when 12v (in your case from the existing horn circuit) is applied to the solenoid. Remember that a solenoid (for the purpose of this conversation) is nothing more than an electrically operated valve, so no voltage gives no airflow and 12v gives full airflow making the horns sound.
Whew...
I apologize if I have totally confused the issue, sometimes I'm not too great with typing out my ideas...
Keep in mind that some kits come with a manual valve like some 18-wheelers have for a roof mounted horn. Sometimes the electric solenoid is an extra cost piece.
Hopefully this helps.
Mike