How can I talk to another user’s systemd user process?

From a script that runs as root, I’d like to start a systemd user service.

I’ve tried the most obvious thing:

sudo -u $ user systemctl --user <etc>

but I get Failed to connect to bus: No such file or directory – I assume this is referring to the dbus system bus.

Do I need to set DBUS_SYSTEM_BUS_ADDRESS? How would I find out what to set it to?

Or is there a different way to interact with another user’s systemd?