Mr. Cheney has become a central figure in his eldest daughter’s campaign. When Ms. Cheney began her bid over the summer, she told associates that she wanted to win in her own right.

But she has plainly recognized what an asset her father can be in a Republican primary. And according to Republicans who have spoken to him, Mr. Cheney, his once-faltering health revived by a new heart, is consumed with helping her win.

He is hosting fund-raisers not just for the new super PAC but also for the campaign itself — he appeared with Ms. Cheney at a finance event in Denver last month.

And Ms. Cheney has not been shy about emphasizing their relationship.