owners handbook says Fuse 36, 20 amp. (some cars are only 15 amp for this circuit) .
Devices that come with a lead and a plug that plugs into the aux socket often have a fuse inside the plug. The plug unscrews. Or there may be a fuse in the device itself. have you checked this fuse?
Some devices such as heavy duty tyre compressor, and possibly inverters , draw too many amps to use an aux plug. They should come with clips to connect direct to the battery terminals. But if some how its drawing close to 20 amps for a sustained period through the aux circuit it might overheat the wiring or fittings without blowing the fuse. Sorry I cant help with how to access it for testing.
I once tried running a fan assisted coolbox through a cheap and nasty splitter that gave me two sockets. (It had a lead and plug with a fuse. The coolbox plug also had a fuse -8 amp IIRC) It melted the splitter wiring without blowing any of the fuses - car,splitter or device) . Could have been nasty.
My solution was to install a second aux socket with its own wire to a spare fuse slot in the fusebox. But even this required precautions to ensure the wiring was heavy duty and as short as possible.
This was a Yaris. I havnt bothered with the jazz. I no longer use the fan on the coolbox. It didnt help that much.