Adjunctions by units and counits

Unit + UMP definition of adjunction

Quoting from Awodey's "Category Theory": An adjunction between categories C and D consists of functors F : C ↔ D : G and a natural transformation T : 1_C → G ∘ F with the property: (o) For any c : C, d : D, and f : c → G d, there exists a unique g : F c → d such that f = (G g) ∘ (T c) as indicated in

                g
     F c ..................> d

                 G g
     G (F c) --------------> G d
       ^                    _
       |                    /|
       |                  /
       |                /
       |              /
       | T c        /
       |          /  f
       |        /
       |      /
       |    /
       |  /
        c

Terminology and notation:

• F is called the left adjoint, G is called the right adjoint, and T is called the unit of the adjunction.
• One sometimes writes F -| G for ``F is left and G right adjoint.''
• The statement (o) is the UMP of the unit T.

Note that the situation F ⊣ G is a generalization of equivalence of categories, in that a pseudo-inverse is an adjoint. In that case, however, it is the relation between categories that one is interested in. Here, one is concerned with the relation between special functors. That is to say, it is not the relation on categories ``there exists an adjunction,'' but rather ``this functor has an adjoint'' that we are concerned with.

Counit + UMP definition of adjunction

Paraphrasing and quoting from Awody's "Category Theory": An adjunction between categories C and D consists of functors F : C ↔ D : G and a natural transformation U : F ∘ G → 1_D with the property: (o) For any c : C, d : D, and g : F c → d, there exists a unique f : c → G d such that g = (U d) ∘ (F f) as indicated in the diagram

                f
     c ..................> G d

               F f
     F c --------------> F (G d)
      \                    |
        \                  |
          \                |
            \              |
              \            | U d
             g  \          |
                  \        |
                    \      |
                      \    |
                       _\| V
                          d

Terminology and notation:

• The statement (o) is the UMP of the counit U.

The counit is just the dual of the unit. We formalize this here so that we can use it to make coercions easier.

Unit + Counit + Zig + Zag definition of adjunction

Quoting Wikipedia on Adjoint Functors: A counit-unit adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and two natural transformations

      ε : FG → 1_C
      η : 1_D → GF

respectively called the counit and the unit of the adjunction (terminology from universal algebra), such that the compositions

          F η            ε F
      F -------> F G F -------> F

          η G            G ε
      G -------> G F G -------> G

are the identity transformations 1_F and 1_G on F and G respectively. In this situation we say that ``F is left adjoint to G'' and ''G is right adjoint to F'', and may indicate this relationship by writing (ε, η) : F ⊣ G, or simply F ⊣ G. In equation form, the above conditions on (ε, η) are the counit-unit equations

      1_F = ε F ∘ F η
      1_G = G ε ∘ η G

which mean that for each X in C and each Y in D,

      1_{FY} = ε_{FY} ∘ F(η_Y)
      1_{GX} = G(ε_X) ∘ η_{GX}

These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the ``zig-zag equations'' because of the appearance of the corresponding string diagrams. A way to remember them is to first write down the nonsensical equation 1 = ε ∘ η and then fill in either F or G in one of the two simple ways which make the compositions defined. Note: The use of the prefix ``co'' in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an initial property whereas the counit morphisms will satisfy terminal properties, and dually. The term unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.

Use the per-object version of the equations, so that we don't need the associator in the middle. Also, explicitly simplify some of the types so that rewrite works better.