Suppose that I have a small coil that is moved at a steady rate into a solenoid. The solenoid already has a current passing through (basically, it is an electromagnet). The small coil starts beyond the solenoid, moves towards it, into it, then out of it through the other end. What is the magnetic flux inside the solenoid? (i.e. What is the shape of the magnetic flux vs. time graph inside the solenoid?)

Once the coil is completely inside the solenoid, the magnetic flux is constant as the magnetic field inside is uniform. However, is the magnetic flux inside the solenoid increasing when the coil approaches, or decreasing? I personally think it is decreasing because the coil induces a field to oppose the field of the solenoid, while the field of the solenoid is unchanging. Hence, the field inside the solenoid is partially cancelled out, and the magnetic flux is decreasing. However, I was told that I am wrong. Can anyone explain why?