majenko:
Just a note on your terminology...A 5V wall wart doesn't have a -5V line.
You have a +(whatever)V connection, and a GND connection. The same goes for the Arduino, the op-amp power supply, etc. You have + and GND, not + and -.
For a 5V supply you have +5V and GND. For a 10V supply, it is +10V and GND.
If you had +5V and -5V, then the difference between them would be 10V! (ouch).
Also, I assume you have a 10V power supply to run the op-amp from? You do realise that you can't power the op-amp from 5V and expect a gain of 2 to give you 10V out of it...?
I don't know if there is any really accurate terminology one can use with DC wall warts that doesn't either make assumptions or needs context to be understood correctly.
Certainly a 'wall wart' type DC voltage source has both a positive and negative terminal (just like a battery) usually terminated into a coaxial DC connector, but no standard as to if center pin is positive or outer shell is positive, they come in both flavours. And one is free to connect either the negative terminal to one's circuit common and that results is a positive 5vdc available to power the circuitry, but of course there is nothing wrong with connecting the positive terminal to one's circuit common when they do require a -5vdc voltage source as for perhaps a op-amp's negative rail. But it's certainly true that a 5 volt wall wart doesn't have both a +5vdc terminal and a -5vdc terminal, at least not at the same time. 
And the term 'ground' is such an overloaded term that can have several meanings and is a major pain in the butt to determine what one really means without the specific context being explained. I prefer the term circuit common or just common, over the term ground. And calling it GND is no help as that is almost impossible to pronounce. But then again I'm a crusty old hardware type that has his idiosyncrasies.
Lefty