I personally have both a Sony a6400 and a really old GoPro (Hero3) that both work with my Atem Mini PRO ISO.
If you have no image, it is probably something simple just overlooked. So I'd check over all the basics to look for "obvious mistakes":
- Does the a6400 have an image on its own screen? ie: You are sure it has power, lens cap off, exposure is reasonable, etc.
- Does the a6400 HDMI out work if plugged directly into your monitor or TV? (Proving a6400 HDMI works)
- If it works on another monitor/tv, you know that piece of the puzzle is ok. Now plug into the Atem. Double check which channel you plugged it into (1-4). Double check that channel is activated and that FTB (Fade to black) is NOT activated).
- As a quick check on the output of the Atem, I often set it to "M/V" (Multiview) instead of "Pgm" (Program) as it will show the multiview screen elements even if all sources are black. Then you should know the Atem output is working with your monitor/TV.
- Also if you have played around with the Atem control software, check that you don't have any weird feature enabled for the upstream or downstream keyers.
- If you used the Atem control software, there is the option to "Clear Startup State" to clear any customizations and then power cycle the device to start fresh with default settings.
Check each piece of your setup one-by-one as above and you should be able to isolate the issue! Could be camera, could be cable from camera to atem, could be atem settings, could be cable from atem to monitor/tv, could be monitor/tv. Must test each piece.