That sounds more like the description of the small versions (600/700 or the older 300/500) and not the 800 (and i suspect the 900). The larger models have actually three temperature controllers for the three zones that are controlled (cockpit, forward and aft cabin).
Never in my lifetime, will Bill Boeing, bring 3 controllers in one 737 to control the temperature. You will show me the 3 controller, I owe You 3 beers
Outgoing from the Mix Manifold You have 2 seperate ducts , one into FWD and one into AFT PAX CABIN, one duct from Pack 1 for the Control Cabin, controlled by seperate temperature bulbs , giving their infos in ONE controller. AUTO MODE is a little bit more "fine tuned" as long we can speak from "fine tuning" on my beloved 737.
Rgds