Your cable will work, but your pin #4 on the 9-pin PC side only needs to be connected to pin 5 on the Fanuc side. Pin 4 on the PC side is an output (DTR) and pin 4 on the Fanuc side is also an output. Connecting two outputs together like this can give unpredictable results. Also, pin 1 on the PC side is an input (DCD), and you've got it tied to the shield ground on the Fanuc (pin 1) which is not doing anything.
Connect pin 4 on the PC side to pin 5 on the Fanuc (don't use pin 4 on the Fanuc)
Disconnect the wire from pin 1 on the PC to pin 1 on the Fanuc
If there is a shield in your cable (I hope so), tie it to pin 1 on the Fanuc side only.
There are many other cable layouts that will work also. The important things are:
Data output pin on PC is tied to data input pin on Fanuc (pin 3 - pin 3)
Data input pin on PC is tied to data output pin on Fanuc (pin 2 - pin 2)
Signal ground on PC is tied to signal ground on Fanuc (pin 5 - pin 7)
Some output (pin 4 or 7 on PC or pin 4 on Fanuc) is tied to CTS signal on Fanuc (5)
Some output (pin 4 or 7 on PC or pin 20 on Fanuc) is tied to pins 6 and 8 on Fanuc
Cable shield is grounded (pin 1 on Fanuc, or to connector shell)
We do it like this:
9 pin ---- 25 pin
2 --------- 2
3 --------- 3
5 --------- 7
7 --------- 5
pins 6, 8, and 20 jumpered together on Fanuc side only
pin 1 on Fanuc tied to cable shield.