Why is it possible for a pilot to "fight" the autopilot? From my (layman's) point of view, it seems logical that either the pilot would be in control or the autopilot would be in control, but not both at the same time.
It's not that one or the other is exclusively controlling the plane. Either (or both) can grab the controls. It's kind of a subtle distinction, but the idea is you don't want the pilot's yoke or stick to ever be physically disengaged from the plane, because that's a mechanism that could fail, leaving the plane uncontrollable by anything. Similar with cruise control on a car: the accelerator physically moves when the computer engages, because it's actually pushing the accelerator. That's what you want, rather than the human needing to suddenly speed up and finding the pedal does nothing when pressed.
And once you accept that, then it's natural that there should be some play before one system disengages to favor the other. You don't want a sneeze to disable your autopilot.
And once you accept that, then it's natural that there should be some play before one system disengages to favor the other. You don't want a sneeze to disable your autopilot.