I'm not sure what projector you have or what TV you actually have.
If your projector is 4K and the TV is 1080p, then your resolution will be limited by the TV to 1080p.
So, the solution is to use a scaler/splitter which will split a HDMI source to two different displays with different resolutions.
Like this:
https://www.amazon.com/HDMI-Splitter-1x2-4K-60Hz/dp/B07WR7KP1B/ref=sr_1_3If both displays support full 4K resolution, then just a basic HDMI splitter is all you need.
Typically, a home theater will have a surround sound receiver and other devices as part of the setup to make everything work, so I'm not entirely sure what your setup is.
If you want to hook up several different sources, then you would want a HDMI switcher as part of the setup as well. At the point when you are talking about all of these devices, I would STRONGLY recommend a decent AV receiver and proper surround sound setup as part of the overall solution.
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