Another fascinating picture you've shared!
This was a fairly common system for building and unloading haystacks - a simple rope and pulley system. There's probably a horse, out of shot, pulling the end of the rope attached to the pulley and thus lifting the hay (which was probably contained within a 'hay sling'). The system was not only quicker and a great deal less effort than using pitchforks, it was also more efficient: you didn't lose hay from every (small) forkload.
I don't know that the apparatus ever enjoyed a specific, unique name, although perhaps another user knows differently. A farmer wanting to use such a system probably constructed his own, using parts from the local agricultural merchant.
Here's a conservation/farm heritage/re-enactment group in Dartmoor using a similar system:
