St 4700 shifter trim down question
It seems that the problem is that the cable is not beign pulled through the hose. this can be caused by various things:

Get your chain on the bigger chainring and shift down, since the derailleur does not shift properly, try to move it by hand. if you feel just a little resistance you should check your routing and/or change hoses and cables. if the resistens is strong, do not apply to much force. In this case check if the limit adjuster screw is preventing the chain from moving onto the smaller chainring. you should find two screws on the derailleur which are responsible for setting the upper and lower bound, e.g. prevent your derailleur from moving the chain to far.

hope that helps!

PS.: make shure that the cable is routed the right way, it looks kind of wrong in the posted picture (near the point where the cable is clamped).