It will adjust down automatically, but it takes a while and goes very slowly. I am not satisfied with the way it does it.
If the lower speed zone has two speed limit signs, the car will ignore the first one. It will register the 2nd one and then after some delay it will slow down to it very slowly. It has always been like this and I don't like it either, so I usually take over and tap on the brake or up on the gear stalk to disable AP or FSD and slow down myself quicker. Another way to try is to roll down the right thumb wheel to set a lower speed, but it also takes forever to slow down to that speed. I wish Tesla would give us the option to slow down to the speed limit right at the sign and not 1/2 mile past it.