GtkWindowunstick

Asks to unstick window, which means that it will appear on only one of the user’s desktops. Note that you shouldn’t assume the window is definitely unstuck afterward, because other entities (e.g. the user or [window manager][gtk-X11-arch]) could stick it again. But normally the window will end up unstuck. Just don’t write code that crashes if not.

You can track stickiness via the “window-state-event” signal on GtkWidget.