Editing Adjoint functors (section)

===Universal morphisms induce hom-set adjunction===

Given a right adjoint functor ''G'' : ''C'' → ''D''; in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps.

* Construct a functor {{itco|''f''}} : ''D'' → ''C'' and a natural transformation ''η''.
** For each object ''Y'' in ''D'', choose an initial morphism ({{itco|''f''}}(''Y''), ''η''<sub>''Y''</sub>) from ''Y'' to ''G'', so that ''η''<sub>''Y''</sub> : ''Y'' → ''G''({{itco|''f''}}(''Y'')). We have the map of {{itco|''f''}} on objects and the family of morphisms ''η''.
** For each {{itco|''f''}} : ''Y''<sub>0</sub> → ''Y''<sub>1</sub>, as ({{itco|''f''}}(''Y''<sub>0</sub>), ''η''<sub>''Y''<sub>0</sub></sub>) is an initial morphism, then factorize ''η''<sub>''Y''<sub>1</sub></sub> &compfn; {{itco|''f''}} with ''η''<sub>''Y''<sub>0</sub></sub> and get {{itco|''f''}}({{itco|''f''}}) : {{itco|''f''}}(''Y''<sub>0</sub>) → {{itco|''f''}}(''Y''<sub>1</sub>). This is the map of {{itco|''f''}} on morphisms.
** The commuting diagram of that factorization implies the commuting diagram of natural transformations, so ''η'' : 1<sub>''D''</sub> → ''G'' &compfn; {{itco|''f''}} is a [[natural transformation]].
** Uniqueness of that factorization and that ''G'' is a functor implies that the map of {{itco|''f''}} on morphisms preserves compositions and identities.
* Construct a natural isomorphism Φ : hom<sub>''C''</sub>({{itco|''f''}}&minus;,&minus;) → hom<sub>''D''</sub>(&minus;,''G''&minus;).
** For each object ''X'' in ''C'', each object ''Y'' in ''D'', as ({{itco|''f''}}(''Y''), ''η''<sub>''Y''</sub>) is an initial morphism, then Φ<sub>''Y'', ''X''</sub> is a bijection, where Φ<sub>''Y'', ''X''</sub>({{itco|''f''}} : {{itco|''f''}}(''Y'') → ''X'') = ''G''({{itco|''f''}}) &compfn; ''η''<sub>''Y''</sub>.
** ''η'' is a natural transformation, ''G'' is a functor, then for any objects ''X''<sub>0</sub>, ''X''<sub>1</sub> in ''C'', any objects ''Y''<sub>0</sub>, ''Y''<sub>1</sub> in ''D'', any ''x'' : ''X''<sub>0</sub> → ''X''<sub>1</sub>, any ''y'' : ''Y''<sub>1</sub> → ''Y''<sub>0</sub>, we have Φ<sub>''Y''<sub>1</sub>, ''X''<sub>1</sub></sub>(''x'' &compfn; {{itco|''f''}} &compfn; {{itco|''f''}}(''y'')) = G(''x'') &compfn; ''G''({{itco|''f''}}) &compfn; ''G''({{itco|''f''}}(''y'')) &compfn; ''η''<sub>''Y''<sub>1</sub></sub> = ''G''(''x'') &compfn; ''G''({{itco|''f''}}) &compfn; ''η''<sub>''Y''<sub>0</sub></sub> &compfn; ''y'' = ''G''(''x'') &compfn; Φ<sub>''Y''<sub>0</sub>, ''X''<sub>0</sub></sub>(&compfn;) &compfn; ''y'', and then Φ is natural in both arguments.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.  (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)