How do I wire a/c if furnace wiring compartment has no y terminal?

I have been trying to figure this out for more than a year. All was running until HVAC company came to replace 2 thermostats with nest and rewired 2 HVAC units (a front of house and a back of house unit)and left me the back unit's a/c not working. I pressed the contactor of the compressor (manually to test) and it runs. Also the green cable gets the 24 volts. Yet, when I turn the nest to a/c only the fan turns on.
The second unit I have is easier for me to understand because there is a y terminal on the furnace board but this one does not have a y terminal/black wire.

I would really, really appreciate if one of you experts helps me with the basic question: Can the yellow/y wire from thermostat go directly to the contactor cable without having been connected to a wire in the furnace wiring system? Nest told me that it should be connected to the furnace wiring for "translation of the signal" but they just do not know how I can do that if the furnace wiring does not have a wire labeled y.
or How can I find out which one of those wired is y? Maybe the little tag with the y fell off? The other thing I have noticed is that the low voltage wires that go to the compressor from this unit and the wires that go to the compressor from the second unit seem to be connected at some point, somewhere because outside, by the 2 compressors there is only one conduit that comes out of the wall and it has only 5 wires as the bottom picture shows. Out of these 5 wires, the yellow and red activate the working unit and the 3 left (green and white and blue) do not have power and thus do not activate the unit I have the a/c not working on.

Thank you so much for any suggestions or help understanding the system.

Furnace of non working a/c has: Blue wire from thermostat is connected to c from furnace and white cable to compressor.

red wire from thermostat is connected to r from furnace. white from thermostat is connected to w from furnace. green from thermostat is connected to g from furnace.