G-STAW
The replies given will I hope have cleared up the confusion but just to try and tie it together:
there are 3 FCCs (Flight Control Computers) which will independently compute from many varied inputs the required outputs to either the AP or FD and as others have said Left FCC=Left A/P or F/D etc.
Only one FCC is required for FD operation and both pilots can select that FCC simultaneously for FD display on their respective ADIs. Normally 2 FCCs are required to engage an AP (for internal cross check purposes but later aeroplanes seem to have a mod allowing single FCC for AP engagement) but no FDs are required for AP engagement.
The FCC outputs to the FDs, postion the FD command bars as targets for the pilot to follow for the selected AFDS mode and this can be any mode not just LNAV/VNAV.
The FCC outputs to the AP position the flight controls to follow the selected AFDS mode. If the FDs happen to be on they will reflect (hopefully) the commands from a different FCC (see later) that a pilot would require if manually flying. If FDs are selected off the AP will still fly the a/c as above.
With some authorities it is not permitted to have the FD and AP fed simultaneously from the same FCC and so the FD command bars bias out of view (no flags) when that FCC is utilised for AP purposes. It is for this reason that some company SOPs will specify, for one example, that the Centre AP for normal use with each pilot's FCC switch selected to Left or Right respectively for FD display and also why there are no FDs displayed when all 3 APs are engaged on multi-AP approaches. Some authorities seem to permit this or don't stipulate one way or the other so yo may see some a/c with the FD bars displayed even when using the same FCC (just a pin change I believe).
I hope this helps rather than confuse.