A monitor is attached to a Win10 box and a second monitor to 16.04. The right monitor is 16.04 and left monitor is Win10.
I am seeking the name of the "class of software" that describes all instances that performs the function. For example, MS-Excel is a member of the spreadsheet class.
Is there a class of software that allows the Win10 keyboard/mouse pair to detect when the user sends the mouse off the right-edge of the monitor to signal intent to send mouse+keyboard inputs to 16.04? These controls signals could be sent through the LAN. Naturally, the reverse from 16.04 to Win10 would be needed to return the KB/Mouse commands back to the Win10 box.
Hopefully, I have explained this concept well: I welcome any improvement.
As Nick mentioned, Synergy is perhaps the best known software for what you ask. It's no longer free, but for $10 is still pretty reasonable.
If it doesn't work for you (or for your budget), then you may find a good alternative on alternativeto.net.
From there, you may find win2vnc useful; it'll let you control extra computers as long as you're willing to run VNC servers on them. The site claims win2vnc is discontinued, but the SourceForge page is still up, and shows an update from a year ago.
My personal experience is with Synergy and x2vnc, but the latter is only useful if your keyboard and mouse are plugged into your linux box, which is the opposite of how you described your setup.
This is probably irrelevant but will mention in case you do not know of it. why not use a keyboard/mouse switch? You plug your keyboard/mouse into it and then run the cables to both boxes. A press of a button and it your input devices are switched to that box. Some have toggle switches, some a button that goes through the connected boxes sequentially; you get the idea.