Unit of an adjunction

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

The unit of the adjunction is a natural transformation

\eta:\mathrm{Id}_{\mathcal C}\Rightarrow G F

characterized as follows: for each object $c\in\mathcal C$ , the component

\eta_c: c \longrightarrow G(Fc)

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

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

Equivalent characterizations

Equivalently, $\eta$ is the transpose of $\mathrm{id}_F$ under the natural isomorphism of hom-bifunctors.

Remarks

The unit $\eta$ and the counit $\varepsilon$ satisfy the triangle identities (see adjoint functors):

\varepsilon_{F c}\circ F(\eta_c)=\mathrm{id}_{F c} \quad\text{and}\quad G(\varepsilon_d)\circ\eta_{Gd}=\mathrm{id}_{Gd}

for all $c\in\mathcal C$ and $d\in\mathcal D$ .

Examples

Free/forgetful (Set–Grp). For $F:\mathbf{Set}\to\mathbf{Grp}$ free group and $U:\mathbf{Grp}\to\mathbf{Set}$ forgetful with $F\dashv U$ , the unit at a set $X$ is the function $\eta_X: X \to U(F(X))$ sending $x\in X$ to the corresponding generator in the underlying set of the free group.

Product–exponential (Set). For the adjunction $(-)\times X \dashv (-)^X$ in $\mathbf{Set}$ , the unit at $A$ is the function $\eta_A: A \to (A\times X)^X, \qquad \eta_A(a)(x)=(a,x).$ It assigns to $a$ the constant-in- $A$ “graph” map $X\to A\times X$ .

Abelianization–inclusion (Grp–Ab). For $\mathrm{ab}:\mathbf{Grp}\to\mathbf{Ab}$ left adjoint to $i:\mathbf{Ab}\hookrightarrow \mathbf{Grp}$ , the unit at a group $G$ is the canonical quotient homomorphism $\eta_G: G \twoheadrightarrow i(\mathrm{ab}(G)) \cong G/[G,G].$