Adjoint functors

We say $F$ is left adjoint to $G$, and write $F \dashv G$, if for every object $c\in\mathcal C$ and $d\in\mathcal D$ there is a bijection of sets

which is natural in $c$ and $d$ (i.e. it is a natural isomorphism of bifunctors $\operatorname{Hom}_{\mathcal D}(F-, -)\cong \operatorname{Hom}_{\mathcal C}(-, G-)$).

Elements of $\operatorname{Hom}$ are morphisms; the naturality condition says that $\Phi$ commutes with pre- and post-composition in the two variables (using composition in $\mathcal C,\mathcal D$).

An adjunction $F\dashv G$ is equivalently specified by:

• a unit $\eta:\mathrm{Id}_{\mathcal C}\Rightarrow G F$, a natural transformation (see unit),
• a counit $\varepsilon:F G\Rightarrow \mathrm{Id}_{\mathcal D}$, a natural transformation (see counit),

such that the triangle identities hold for all $c\in\mathcal C$, $d\in\mathcal D$:

(Here $\mathrm{id}$ is the identity morphism.)

If moreover $\eta$ and $\varepsilon$ are natural isomorphisms, then $F$ and $G$ form an equivalence of categories.

1. Free/forgetful (Set–Grp). The free group functor $F:\mathbf{Set}\to\mathbf{Grp}$ is left adjoint to the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$. Concretely,
   $$\operatorname{Hom}_{\mathbf{Grp}}(F(X),\,H)\cong \operatorname{Hom}_{\mathbf{Set}}(X,\,U(H)),$$
   naturally in a set $X$ and a group $H$.

1. Product–exponential (Set). Fix a set $X$. In $\mathbf{Set}$, the functor $(-)\times X$ is left adjoint to the exponential functor $(-)^X=\operatorname{Hom}(X,-)$:
   $$\operatorname{Hom}(A\times X,\,B)\cong \operatorname{Hom}(A,\,B^X),$$
   naturally in $A,B\in\mathbf{Set}$. (This is the usual currying bijection.)

1. Abelianization–inclusion (Grp–Ab). Let $i:\mathbf{Ab}\hookrightarrow \mathbf{Grp}$ be the inclusion. The abelianization functor $\mathrm{ab}:\mathbf{Grp}\to\mathbf{Ab}$, $G\mapsto G/[G,G]$, is left adjoint to $i$:
   $$\operatorname{Hom}_{\mathbf{Ab}}(\mathrm{ab}(G),\,A)\cong \operatorname{Hom}_{\mathbf{Grp}}(G,\,i(A)).$$