CounitNatürlichen

In category theory, the counit of an adjunction, sometimes called the natural counit, is a natural transformation that connects the composite of an adjoint pair with the identity on the target category. If F: C → D is left adjoint to G: D → C, then there is a unit η: Id_C → G F and a counit ε: F G → Id_D. The components of ε are ε_Y: F(GY) → Y for each object Y in D.

Naturality means that for every morphism f: Y → Z in D, ε_Z ∘ F(G f) = f ∘ ε_Y. The pair (η, ε) must satisfy the triangle identities, which express coherence between the unit and the counit. Concretely, for all X in C, ε_{F X} ∘ F η_X = Id_{F X}, and for all Y in D, G ε_Y ∘ η_{G Y} = Id_{G Y}. These identities ensure that FG behaves like an idempotent up to the adjunction, while the adjunction itself encodes a precise form of universal mapping property.

Examples help illustrate the concept. A standard one is the free-forgetful adjunction between Set and Mon: the free monoid functor F: Set → Mon is left adjoint to the forgetful functor U: Mon → Set. The counit ε_M: F(U M) → M sends a formal word over the underlying set of M to its evaluation in M, yielding a monoid homomorphism. Another example is the free vector space functor F: Set → Vect_k left adjoint to the forgetful functor U: Vect_k → Set; the counit ε_V: F(U V) → V maps a basis element corresponding to v ∈ V to v.

The counit, together with the unit, also induces a monad on C: the composite GF carries a multiplication μ = G ε F, making (GF, η, μ) a monad.