  Everyone knows that if you have a bivariant homology theory satisfying a base
change formula, you get an representation of a category of correspondences. For
theories in which the covariant and contravariant transfer maps are in mutual
adjunction, these data are actually equivalent. In other words, a 2-category of
correspondences is the universal way to attach to a given 1-category a set of
right adjoints that satisfy a base change formula.
  Through a bivariant version of the Yoneda paradigm, I give a definition of
correspondences in higher category theory and prove an extension theorem for
bivariant functors. Moreover, conditioned on the existence of a 2-dimensional
Grothendieck construction, I provide a proof of the aforementioned universal
property. The methods, morally speaking, employ the `internal logic' of higher
category theory: they make no explicit use of any particular model.
