Suppose there are two pulleys separated by $2c$ distance and connected by a chain; now if we rotate the pulley on one side, will the pulley on the other side rotate instantly? If not, then how will we be able to rotate it in the first place since they are tightly connected by the chain, and as per the theory of relativity, they should not rotate at the same time since information can't travel faster than the speed of light?
$c$ is the distance travelled by the light in one second.
Pressure waves in material objects, including transmitted forces, travel at the speed of sound in the object, which is based upon the density and rigidity of the material.
A material with one of the highest speeds of sound is diamond, at about 12,000 m/s, while most other solids are a few thousand m/s. So any forces in material objects will travel at most this fast, which is far slower than the speed of light. The far end of your pulley system will not "know" anything about forces imparted to the far end until the pressure waves traverse the distance through the rope.
A good demonstration of this is with a slinky, which due to how loose and "unrigid" it is, has an effective speed of sound of only a few m/s, which is visible to the eye.
