Assuming the north pole of the magnet is pointing downwards inside the solenoid, the magnetic flux will be at its maximum downwards initially. However when you move the coil down, the direction of the magnetic flux would still be downwards but with a lower magnitude of flux than initially as it changes. Therefore the change in magnetic flux would be upwards, and according to Lenz's law, the induced current/emf will have a magnetic field such that it opposes the change in flux, creating a downwards magnetic field.
However, moving a magnet down with the coil still is the opposite of the previous situation. The initial magnetic flux will be downwards. As the magnet approaches the centre of the solenoid, the magnitude of the downwards flux increases, therefore there is a downward change in flux. A current/emf will then be induced such that its magnetic field opposes the direction of the change in flux that caused it, i.e (upwards)
In both situations, an emf is induced but in different directions.