The upward force on the rope is am where a is the acceleration of the rope and crate upwards and m is the mass of the crate (the mass of the rope is negligible).
The downward force on the rope is mg where g is the acceleration due to gravity.
The force T (tension) that the rope exerts equally on the crate and the mass moving the rope is then equal to the sum of the forces acting on it,
T = m(a+g)
The force F that the crate exerts on the rope is equal to mg
If the crate isn't moving, a = 0 and so T = F
If the crate is moving upwwards then a > 0 and T > F