Switch – Wiring a 4-way switch

My guess is (and this is only a guess pending more information):

The 3 wire cable containing 'A' goes to the light(s). 'A's associated black wire is unused and its white wire 'B' provides neutral for the light(s). The travelers' associated black wire is wirenuted to the feed wire and connects to the other 3-way sw (you labeled it 'C') on its common screw. The travelers and 'A' connect to the 3-way sw with 'A' on the common screw. The wires 'B' and Neutral wirenut together to provide neutral to the light(s). And finally because a conventional sw doesn't need power, get rid of the pigtail you labeled as 'Hot', while keeping the 2 black wires wirenuted together.

The more information I mentioned is this: It looks like there're 2 3-wire cables coming into the top of the box and 1 2-wire cable coming into the bottom of the box. label the wires, including that black wire that looks like it is shoved into the box unused, as to which cable they're associated with, e.g. cable #1, cable #2 and cable #3. Because cables and wires have two ends, knowing where the other end of a wire is can tell you more than what a tester or meter can.

1. Power off.
2. Pick any 2 of the capped wires formerly attached to the 3-way.
3. Nut those two together.
4. Leave the third wire nutted independently.
5. Power on.
6. Throw any other switch in the 3/4 way group. Does the light turn on or off?
   • all other 3/4-ways work correctly -- fail*. Pick a different pair and go to 1.
   • lights do not work in either position -- success. The two nutted together are the messengers.

Mark both messengers with yellow tape (or other appropriate marking) -- messengers are interchangeable, so you don't need to distinguish them from each other.

The two messengers go on the brass screws of the 3-way switch.

The remaining wire is common and goes on the black screw.

* Or, if you just want to eliminate that 3-way, you can stop right here and you are done.