Within a quadra, people have the same priorities in life, the same goals, the same ways of looking at human relationships.
Alpha: they want to enjoy the exploration of coherent ideas in a cozy emotional and sensorial atmosphere.
Beta: they want power, and admiration from people generally, while following a consistent path in the longer term in a ideologically-consistent world.
Gamma: we want to follow a path consistent in the longer term with material prosperity, efficiency, and close, constant emotional bonding with selected individuals, never deviating from our own ethical principles.
Delta: they want to enjoy material comfort and sensory well-being while exploring and maximizing the potential of other individuals.
These broad goals are the same for all types within a quadra. What is different for the types are the "tools" with which each type feels most confident in achieving those goals.
So in Gamma, for instance, let us take the INTp-ESFp dual pair.
Both the ESFp and the INTp want to achieve material property in the long term while remaining true to themselves ethically and through bonding with people. The ESFp admires and needs the INTp's "tools" in seeing the longer-term implications of present actions (otherwise the ESFp isn't sure that they're on the right track) as well as the evaluation of what's efficient and what isn't; the INTp admires and needs the ESFp's tools in terms of working hard towards those goals by connecting to people.
Ideally, your dual is someone you respect the most: they share the same goals as you, but they also seem "better" in being skilled in "tools" that you value but feel yourself sort of lacking in.