66 Dart heater and air conditioning

That is the water control valve. The vacuum dashpot turns a gate valve to turn the water on and off. On for heat, off for AC. The dashpot has a hose connection on each side, the dash control switches the vacuum to the desired port. There is a third hose that feeds the switch, it is fed from the intake manifold port. It typically goes thru the firewall in a grommet, then inside to the switch. I think it is inbetween the master cylinder and distributor in your picture. The vacuum hoses to the water control dashpot come thru the firewall near fan motor on passenger side firewall.

The AC control switch is often brittle, may leak, and need care or replacement. It controls both electrical and vacuum functions, there is also a vacuum dashpot on heater evaporator box.