Adjoint functors

Adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. By definition, an adjunction between categories C and D is a pair of functors

F : D \to C and G : C \to D

and, for all objects $X$ in $C$ and $Y$ in $D$ a bijection between the respective morphism sets

{hom}_{C} (F Y, X) ≅ {hom}_{D} (Y, G X)

such that this family of bijections is natural in $X$ and $Y$ . For more info (the meaning of natural, here, for example), see the wikipedia page Adjoint functors.

The left part of a pair of adjoint functors is one of two best approximations to a weak inverse of the other functor of the pair. (The other best approximation is the functor's right adjoint, if it exists. ) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an adjoint equivalence.

A left adjoint to a forgetful functor is called a free functor; in general, left adjoints may be thought of as being defined freely, consisting of anything that an inverse might require, regardless of whether it works...

Concrete example

Let:

$D = Set$ (category of sets and functions)
$C = Grp$ (category of groups and group homomorphisms)
$G : Grp \to Set$ be the forgetful functor that takes a group to its underlying set (forgets the group operation).
$F : Set \to Grp$ be the free group functor that takes a set $S$ to the free group $F (S)$ generated by $S$ .

Then $F$ is left adjoint to $G$ , written $F ⊣ G$ .

What does the adjunction mean concretely?

The adjunction says there is a natural bijection:

{hom}_{Grp} (F (S), H) ≅ {hom}_{Set} (S, G (H))

where:

Left side: group homomorphisms from the free group on $S$ to a group $H$ .
Right side: functions from the set $S$ to the underlying set of $H$ .

This makes sense:
To give a group homomorphism $F (S) \to H$ , you just need to decide where each generator (element of $S$ ) goes in $H$ , because in a free group there are no relations to preserve besides those forced by group axioms. That choice is exactly a function $S \to G (H)$ .

Conversely, any function from $S$ to the set $G (H)$ extends uniquely to a group homomorphism from the free group on $S$ to $H$ .

Naturality

Naturality here means this correspondence works consistently when you change $S$ or $H$ via morphisms.
If you have a function $S \to S^{'}$ , it induces a map $F (S) \to F (S^{'})$ (by functoriality of $F$ ), and the bijection respects that.
Similarly, a group homomorphism $H \to H^{'}$ induces a function $G (H) \to G (H^{'})$ , and again the correspondence behaves well.