I have a Switch 3. If I just plugged it in to my H9, what would it do? Is there a way to use all three switches? How does that work?

As lordradish said, use a stereo cable and connect the Aux switch to the Exp jack on the back of the H9. Make sure that in H9 Control under the Pedal tab, General Settings you have the “Use Expression Pedal, Aux Switch, or Both” setting set to “Aux Switch” and then on the Pedal tab click “Aux Switch Mappings” to control what each button does. It takes a little bit of work to get things set the way you want them, but I haven’t found it to be a big problem. 

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But I’m already using an expression pedal, and would like to keep doing so. Does that mean I can’t use the Switch 3 with just the aux?

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Oh, that’s a problem.  I don’t think that’s possible.  I think the most you can do is 2 switches with the OX9 and add expression to that.