Hey there..
I have changed the water pump to my 5.0s / c, now I'm a little unsure if all the air is out of the system, especially over the compressor, how do you air it out?
Is this for cooling the compressor or the inlet air ?

Thanks, its a XKR 5.0 from 2010

You really need to vacuum fill the coolant on these cars. I did mine the old way, relying on gravity but the IAT always read higher than usual. I bought a vacuum bleeder and bled it, IATs dropped more than 25 degrees after vacuum bleeding.
If you don’t have the tool, disconnect the two pipes on each side of the charge cooler and leave them disconnected until coolant starts to flow from them, then reconnect. Start the engine and let it run till the thermostat opens. Keep checking the level and top up as necessary.
‘The vacuum bleeders are around $80 on Amazon for a decent one and well worth the money in my opinion.

Out of interest what were the before and after temperatures and what was the corresponding ambient temperature?

Pull one of those hoses off and start sucking! My brother towed his in when he did the same thing, and couldnt get the air out😁.