Adjoint functors
Definition via counit–unit adjunction

A counit–unit adjunction between two categories C and D consists of two functors F : D → C and G : C → D and two natural transformations

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

are the identity transformations 1 and 1 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 {\displaystyle (\varepsilon,\eta):F\dashv G} , or simply F ⊣ G {\displaystyle F\dashv G}.

In equation form, the above conditions on (ε, η) are the counit–unit equations

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

Note that 1 C {\displaystyle 1_{\mathcal {C}}} denotes the identify functor on the category C {\displaystyle {\mathcal {C}}}, 1 F {\displaystyle 1_{F}} denotes the identity natural transformation from the functor F to itself, and 1 F Y {\displaystyle 1_{FY}} denotes the identity morphism of the object FY.

These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the triangle identities, or sometimes 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 {\displaystyle 1=\varepsilon \circ \eta } 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.
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