Adjoint Functors - Formal Definitions

Formal Definitions

There are various definitions for adjoint functors. Their equivalence is elementary but not at all trivial and in fact highly useful. This article provides several such definitions:

• The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.
• The definition via counit-unit adjunction is convenient for proofs about functors which are known to be adjoint, because they provide formulas that can be directly manipulated.
• The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.

Adjoint functors arise everywhere, in all areas of mathematics. Their full usefulness lies in that the structure in any of these definitions gives rise to the structures in the others via a long but trivial series of deductions. Thus, switching between them makes implicit use of a great deal of tedious details that would otherwise have to be repeated separately in every subject area. For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits.