USB - HDMI adapter seems to not work

[wanton cedar]

Check the "USB Type-C Alternate Mode drivers " section of your kernel config:

> Device Drivers                                                              │                                                          │
│                                                          │       -> USB support (USB_SUPPORT [=y])                                            │                                                          │
│                                                          │         -> USB Type-C Support (TYPEC [=m])   

Another easy way to find out if your kernel config is missing something is to try to run with the distribution kernel

Not all USB ports support HDMI/DP alt mode, even if it is a type-C one, so it could be that this simply will not work for your hardware.

[wanton cedar]

the adaptor being usb alt mode capable is one thing, the port you plug it in must be too

if the graphics card is not wired up to communicate via the usb interface then it simply cannot work

there is no software required for this to work if the kernel is capable

it should show up as yet another screen in xrandr/wayland, if you have two graphics cards and are using X then you might have to use the --setprovideroutputsource option

this is usually wired through the integrated GPU

Try xrandr --listproviders to find out if there is a second provider that can be configured as secondary GPU