Counit of an adjunction

For F ⊣ G, the counit ε: F∘G ⇒ Id_D is the natural transformation corresponding to identities under the adjunction bijection.

Counit of an adjunction

Let $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal C$ be functors with an adjunction $F\dashv G$ .

Definition (Counit)

The counit of the adjunction is a natural transformation

\varepsilon: F G \Rightarrow \mathrm{Id}_{\mathcal D}

characterized as follows: for each object $d\in\mathcal D$ , the component

\varepsilon_d: F(Gd) \longrightarrow d

is the unique morphism corresponding to the identity $\mathrm{id}_{G d}\in\operatorname{Hom}_{\mathcal C}(G d, G d)$ under the adjunction bijection

\operatorname{Hom}_{\mathcal D}(F c,\, d)\cong \operatorname{Hom}_{\mathcal C}(c,\, G d).

Equivalently, $\varepsilon$ is the transpose of $\mathrm{id}_G$ .

The counit $\varepsilon$ and the unit $\eta$ satisfy the triangle identities:

G(\varepsilon_d)\circ \eta_{G d}=\mathrm{id}_{G d} \quad\text{for all }d\in\mathcal D.

Free/forgetful (Set–Grp). For $F\dashv U$ (free group and forgetful), the counit at a group $H$ is the homomorphism
$\varepsilon_H: F(U(H)) \to H$
sending a formal word in the underlying set $U(H)$ to its evaluation in $H$ .
Product–exponential (Set). For $(-)\times X \dashv (-)^X$ in $\mathbf{Set}$ , the counit at $B$ is the evaluation map
$\varepsilon_B: B^X \times X \to B, \qquad \varepsilon_B(f,x)=f(x).$
Abelianization–inclusion (Grp–Ab). For $\mathrm{ab}\dashv i$ , the counit at an abelian group $A$ is (canonically) the identity isomorphism
$\varepsilon_A: \mathrm{ab}(i(A)) \xrightarrow{\ \cong\ } A,$
since the abelianization of an already abelian group is itself.