For these stars it never gets hot enough in the core to have Carbon or Oxygen fusion to occur. We are seeing the core of the AGB star composed of Carbon, Oxygen with a small layer of Hydrogen and Helium. Once the PN envelope has expanded away an dissipated the AGB core sits at the top of the White Dwarf sequence in the H-R Diagram. The core is hot and therefore blue and small with a correspondingly low luminosity.
White dwarfs are alot like hot embers.
As they radiate away energy, they lose energy and with no energy source to replace it, they cool off. You might expect the WDs to shrink as they cool, but because they are supported by degeneracy, they don't. They just get cooler and cooler and fainter and fainter till they are essentially lost to observations. As they cool, there is a point at which the Carbon atoms form a lattice with the still roaming around in their own quantum phase space. Because the Galaxy has a finite age, the oldest WDs have not cooled to invisibility yet.