He 'campaigned for president/governor/mayor'. = He 'campaigned' for the presidency/governorship/mayoralty. [=he tried to be elected president/governor/mayor]
The group launched/mounted/led a 'campaign' to protect the area from commercial development.
The university is organizing a 'campaign' to attract a more diverse student population.
(..) You said she would be here at ten this morning. It is now past five. - She's not at the hotel. - Forget about her. We've started the campaign. There isn't time to get someone else. Well, where is she? Maybe at the top of the Eiffel Tower, or the bottom of the Seine. Maybe she's in a traffic jam. (..)1957 Funny Face