Relay wiring
You may need a diode to allow two separate power supplies to control a single output.
Here is the ones I use off amazon.
12vdc amperage unknown but I’ve run 20amp at 12vdc thru multiple times with no burn out.
Hopkins 48955 Towed Vehicle Diodes Kit
https://www.amazon.com/dp/B0002Q81Y4/?tag=joeychgo-20