Family installed a new Nest thermostat without photos of old wiring. Please help rewire 4-wire system to restore AC functionality!

Question

Photo Album of current furnace, setup, and wiring: https://imgur.com/a/VYa2MnT
Hi guys, I'm in town visiting my family for the summer. They installed a Nest thermostat without taking any photos of the pre-existing setup, and have wired it using a guide online. Currently, the fan and heat work fine, and so does the Nest itself, but there's no AC at all when it's pretty clear that there is AC installed in the home.
The home has AC as well as a heating furnace, but I couldn't locate a Y wire to connect to the thermostat. A long time ago before we moved in, the old thermostat (also in the album) was completely inoperable without batteries installed, which led me to believe that this system has no C wire, and should be wired as follows:

Rc - Red

Y - Blue

G - Green

W - White

Would this be a valid config? I'd like to check to see if the blue wire is my cooling wire, but I don't see any way of accessing the hat that's caulked in place on top of my furnace.
Any help would be appreciated! With summer slowly starting to roll around this would be something I'd like to help taken care of before it gets too toasty.

Harper - Reinstate Monica · Answer

OK, so the way thermostat cable is made, there are certain wires in it.

Red
White
Green is added in /3 thermostat cable
Yellow is added in /4 cable
Blue is added in /5 cable

Separate from that, there are standard colors for thermostat wiring.   It starts with a basic heat-only system, then adds ability to control/override fan, then adds ability to control A/C.  And here's how that goes.

"R" is the "supply" wire for 24V control power. The thermostat connects these to various other wires to make calls for heat, fan, etc.
"W" is the "call for heat" wire.  When this is shorted to R, that tells the furnace to fire up and make heat (however it does that). If the furnace has a forced-air blower, the furnace is expected to sequence that all by itself.
"G" is the "call for fan" wire.  When this is shorted to R, tells the furnace to run the blower even if it wouldn't otherwise.
"Y" is the "Call for A/C" wire.  (Shorting it to R says A/C is desired.

You may have noticed that each of those letters is also the first letter of a color.  And lo and behold, they all match thermostat cable, even the ascending feature-set!  Isn't that clever?
Now, the 5th wire is blue, and it doesn't really have an assignment in a conventional furnace+AC system.
Enter the requirement for smart thermostats that need power of their own.  They want the "C" wire, which is the other terminal on the transformer from "R".  There isn't really a convention for what color to use, and obviously no primary colors start with C.  However, since the 5th wire is typically unused and installed as a spare, using blue for "C" has become a convention.
So, I think, now you know how to hook up a conventional thermostat.  Note that you should expect R W Y and G to be connected at the furnace... but generally the C line to the thermostat is NOT connected at the furnace end, and you'll need to consult furnace documentation to figure out where to attach it.  C actually is distributed to the system relays and contactors, to complete the circuit.
(transformer via R to thermostat, via Y to A/C contactor, and via C back to transformer).
C doesn't go anywhere near the thermostat normally.

Family installed a new Nest thermostat without photos of old wiring. Please help rewire 4-wire system to restore AC functionality!

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