I cannot speak about the Airbus ,but a simple answer to Cat 2 and Cat 3 is Cat 2 the a/pilot will controle in all axis as a normal a/p and will follow the Flight Management System ,compass demands,plus speed via auto throttle and radio demands ie glide slope and ILS but cannot be used to land the plane automaticly in auto pilot A OR B but for full Autoland BOTH autopilot channels must be engaged to land the a/c "Hands Off" in bad visability company policy dictates minimum visibility . they all work should 1 channel disagree with the others and on modern a/c there are 4 the odd one out will automaticly be disconnected and the system will be down graded. Must admit it has been quite a while since I worked on the systems but imagine it is the same now with more bits added
