Deal with media holding business up (7)
I believe the answer is:
process
'deal with' is the definition.
(processing is a kind of dealing with)
'media holding business up' is the wordplay.
'media' becomes 'press' (the press is part of the media).
'holding' indicates putting letters inside.
'business' becomes 'co' (short for company).
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'co' backwards is 'oc'.
'press' enclosing 'oc' is 'PROCESS'.
(Other definitions for process that I've seen before include "Series of actions with a single purpose" , "move in train" , "Procedure" , "Method of production" , "Series of actions to achieve a result" .)