How do I set up the mouse to work with two different computers?

Download Logitech Options on both systems and make sure the mouse is ON. Long press the Easy-Switch button to enter pairing mode, connect to the first computer via Unifying USB receiver or Bluetooth, then short press Easy-Switch to switch to the next channel, long press again to enter pairing mode, and connect to the second computer. Enable FLOW through Logitech Options or press Easy-Switch to switch between devices. (Page 6)

From the official Logitech M585 M590 Silent Wireless Mouse manual, page 6

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