With X11 forwarding in ssh, the environment variable that is automatically set in the remote host shell is DISPLAY. This variable specifies the name of the X display to which X11 clients should connect. When X11 forwarding is enabled, the ssh server sets the DISPLAY variable to a value like localhost:10.0, which means that the X11 clients will connect to a proxy X11 display on the remote host. The proxy display will then forward the X11 protocol over ssh to the X server on the local host. This way, the X11 clients can display their graphical output on the local host, even though they are running on the remote host. If X11 forwarding is not enabled, the DISPLAY variable is not set by the ssh server, and the X11 clients will not be able to connect to any X display unless the user manually sets the DISPLAY variable to a valid value. However, this may not work if the X server on the local host does not allow remote connections or if there are firewall rules that block the X11 traffic.

References:

3: Built-in SSH X11 forwarding in PowerShell or Windows Command Prompt - X410

4: Understanding X11 Forwarding through SSH - start to finish steps

1: Why use ssh X11 forwarding with LSF; How to use ssh X11 forwarding - IBM