This article explains how to use a PS5 DualSense controller on PS4. Bluetooth Adapters, PS4, and DualSense The PS5 controller doesn't just work with the PS4, so you must pick up a Bluetooth adapter.

How to use a PS5 controller on PS4 If you want to play your PS4 with a PS5 controller for free, you will need to use the PS Remote Play app on a computer or smartphone.

Learn how to connect your PS4 or PS5 controller to your PC, to determine which games it works with, and to make it work with titles from other platforms.