Editing Adjoint functors (section)

===Free groups===
The construction of [[free group]]s is a common and illuminating example.

Let ''F'' : '''[[category of sets|Set]]''' → '''[[category of groups|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 <math>\eta_Y:Y\to GFY</math> 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 <math>\eta_Y:Y\to GFY</math> via a unique group homomorphism from ''FY'' to ''W''.  This is precisely the [[Free group#Universal property|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 <math>\varepsilon_X:FGX\to X</math> be the group homomorphism that sends the generators of ''FGX'' to the elements of ''X'' they correspond to, which exists by the universal property of free groups. Then each <math>(GX,\varepsilon_X)</math> is a terminal morphism from ''F'' to ''X'', because any group homomorphism from a free group ''FZ'' to ''X'' will factor through <math>\varepsilon_X:FGX\to X</math> 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 <math>(\varepsilon,\eta):F\dashv G</math> is as follows:

; The first counit–unit equation : <math>1_F = \varepsilon F\circ F\eta</math> says that for each set ''Y'' the composition
::<math>FY\xrightarrow\overset{}{\;F(\eta_Y)\;}FGFY\xrightarrow{\;\varepsilon_{FY}\,}FY</math>
: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 <math>F(\eta_Y)</math> 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 <math>\varepsilon_{FY}</math> 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 : <math>1_G = G\varepsilon \circ \eta G</math> says that for each group ''X'' the composition
::<math>GX\xrightarrow{\;\eta_{GX}\;}GFGX\xrightarrow\overset{}{\;G(\varepsilon_X)\,}GX</math>&nbsp;
:should be the identity.  The intermediate set ''GFGX'' is just the underlying set of ''FGX''.  The arrow <math>\eta_{GX}</math> is the "inclusion of generators" set map from the set ''GX'' to the set ''GFGX''.  The arrow <math>G(\varepsilon_X)</math> 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''.