Adapters that allow one to capture an HDMI video signal can be used for a variety of purposes, recording game footage included. They typically cost from a few dozen to around a hundred (BestBuy link) or two hundred US dollars; the product discussed here in this little article, on the other hand, can be yours for a fraction of the price (Aliexpress link).
As reviewed by the MySKU forum user ZachemKupil, the adapter has no prominent branding on its casing and is fairly straightforward to use, as expected from modern plug-and-play devices.
No particular setting up procedure is required; at the very least, the MySKU post does not mention one. The adapter is said to work well enough mated to the OBS Studio software. One thing worth mentioning is that, despite the product supposedly being available in a USB 3.0 and a USB 2.0 version, both are actually USB 2.0-spec; it's just the connector that's different.
1080p video capture is confirmed to work without issues with 2160p30 input supported as well according to the spec sheet on Aliexpress. Please note that delivery times tend to be anywhere from 2 to 4 weeks when ordering inexpensive products from China via platforms such as Aliexpress.
Source(s)
MySKU.club (in Russian)