information:classification, structural:expansion, linguistic:translation
Free groups

The construction of free groups is a common and illuminating example.

Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G:

Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let η Y: Y → G F Y {\displaystyle \eta _{Y}:Y\to GFY} be the set map given by "inclusion of generators". This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through η Y: Y → G F Y {\displaystyle \eta _{Y}:Y\to GFY} via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y.

Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let ε X: F G X → X {\displaystyle \varepsilon _{X}:FGX\to X} be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each (G X, ε X) {\displaystyle (GX,\varepsilon _{X})} is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through ε X: F G X → X {\displaystyle \varepsilon _{X}:FGX\to X} via a unique set map from Z to GX. This means that (F, G) is an adjoint pair.

Hom-set adjunction. Group homomorphisms from the free group FY to a group X correspond precisely to maps from the set Y to the set GX: each homomorphism from FY to X is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F, G).

counit–unit adjunction. One can also verify directly that ε and η are natural. Then, a direct verification that they form a counit–unit adjunction (ε, η): F ⊣ G {\displaystyle (\varepsilon,\eta):F\dashv G} is as follows:

The first counit–unit equation 1 F = ε F ∘ F η {\displaystyle 1_{F}=\varepsilon F\circ F\eta } says that for each set Y the composition

should be the identity. The intermediate group FGFY is the free group generated freely by the words of the free group FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow F (η Y) {\displaystyle F(\eta _{Y})} is the group homomorphism from FY into FGFY sending each generator y of FY to the corresponding word of length one (y) as a generator of FGFY. The arrow ε F Y {\displaystyle \varepsilon _{FY}} is the group homomorphism from FGFY to FY sending each generator to the word of FY it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on FY.

The second counit–unit equation 1 G = G ε ∘ η G {\displaystyle 1_{G}=G\varepsilon \circ \eta G} says that for each group X the composition

should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow η G X {\displaystyle \eta _{GX}} is the "inclusion of generators" set map from the set GX to the set GFGX. The arrow G (ε X) {\displaystyle G(\varepsilon _{X})} is the set map from GFGX to GX which underlies the group homomorphism sending each generator of FGX to the element of X it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on GX.