  Category theory has foundational importance because it provides conceptual
lenses to characterize what is important in mathematics. Originally the main
lenses were universal mapping properties and natural transformations. In recent
decades, the notion of adjoint functors has moved to center-stage as category
theory's primary tool to characterize what is important in mathematics. Our
focus here is to present a theory of adjoint functors. The basis for the theory
is laid by first showing that the object-to-object "heteromorphisms" between
the objects of different categories (e.g., insertion of generators as a set to
group map) can be rigorously treated within category theory. The heteromorphic
theory shows that all adjunctions arise from the birepresentations of the
heteromorphisms between the objects of different categories.
