Editing Adjoint functors (section)

==Adjunctions in full==

There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

An ''adjunction'' between categories ''C'' and ''D'' consists of
*A [[functor]] ''F'' : ''D'' → ''C'' called the '''left adjoint'''
*A functor ''G'' : ''C'' → ''D'' called the '''right adjoint'''
*A [[natural isomorphism]] Φ : hom<sub>''C''</sub>(''F''–,–) → hom<sub>''D''</sub>(–,''G''–)
*A [[natural transformation]] ε : ''FG'' → 1<sub>''C''</sub> called the '''counit'''
*A natural transformation η : 1<sub>''D''</sub> → ''GF'' called the '''unit'''

An equivalent formulation, where ''X'' denotes any object of ''C'' and ''Y'' denotes any object of ''D'', is as follows:

::For every ''C''-morphism ''f'' : ''FY'' → ''X'', there is a unique ''D''-morphism Φ<sub>''Y'', ''X''</sub>(''f'') = ''g'' : ''Y'' → ''GX'' such that the diagrams below commute, and for every ''D''-morphism ''g'' : ''Y'' → ''GX'', there is a unique ''C''-morphism Φ<sup>−1</sup><sub>''Y'', ''X''</sub>(''g'') = ''f'' : ''FY'' → ''X'' in ''C'' such that the diagrams below commute:

[[File:Adjoint functors sym.svg|center|350px]]

From this assertion, one can recover that:
*The transformations ε, η, and Φ are related by the equations
:<math>\begin{align}
f = \Phi_{Y,X}^{-1}(g) &= \varepsilon_X\circ F(g) & \in & \, \, \mathrm{hom}_C(F(Y),X)\\
g = \Phi_{Y,X}(f) &= G(f)\circ \eta_Y & \in & \, \, \mathrm{hom}_D(Y,G(X))\\
\Phi_{GX,X}^{-1}(1_{GX}) &= \varepsilon_X  & \in & \, \, \mathrm{hom}_C(FG(X),X)\\
\Phi_{Y,FY}(1_{FY}) &= \eta_Y & \in & \, \, \mathrm{hom}_D(Y,GF(Y))\\
\end{align}
</math>
*The transformations ε, η satisfy the counit–unit equations
:<math>\begin{align}
1_{FY} &= \varepsilon_{FY} \circ F(\eta_Y)\\
1_{GX} &= G(\varepsilon_X) \circ \eta_{GX}
\end{align}</math>
*Each pair (''GX'', ε<sub>''X''</sub>) is a [[universal morphism|terminal morphism]] from ''F'' to ''X'' in ''C''
*Each pair (''FY'', η<sub>''Y''</sub>) is an [[universal morphism|initial morphism]] from ''Y'' to ''G'' in ''D''

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors ''F'' and ''G'' alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.

===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.)

===counit–unit adjunction induces hom-set adjunction===

Given functors ''F'' : ''D'' → ''C'', ''G'' : ''C'' → ''D'', and a counit–unit adjunction (ε, η) : ''F'' &dashv; ''G'', we can construct a hom-set adjunction by finding the natural transformation Φ : hom<sub>''C''</sub>(''F''&minus;,&minus;) → hom<sub>''D''</sub>(&minus;,''G''&minus;) in the following steps:

*For each ''f'' : ''FY'' → ''X'' and each ''g'' : ''Y'' → ''GX'', define
:<math>\begin{align}\Phi_{Y,X}(f) = G(f)\circ \eta_Y\\
\Psi_{Y,X}(g) = \varepsilon_X\circ F(g)\end{align}</math>
:The transformations Φ and Ψ are natural because η and ε are natural.

*Using, in order, that ''F'' is a functor, that ε is natural, and the counit–unit equation 1<sub>''FY''</sub> = ε<sub>''FY''</sub> &compfn; ''F''(η<sub>''Y''</sub>), we obtain
:<math>\begin{align}
\Psi\Phi f &= \varepsilon_X\circ FG(f)\circ F(\eta_Y) \\
 &= f\circ \varepsilon_{FY}\circ F(\eta_Y) \\
 &= f\circ 1_{FY} = f\end{align}</math>
:hence ΨΦ is the identity transformation.

*Dually, using that ''G'' is a functor, that η is natural, and the counit–unit equation 1<sub>''GX''</sub> = ''G''(ε<sub>''X''</sub>) &compfn; η<sub>''GX''</sub>, we obtain
:<math>\begin{align}
\Phi\Psi g &= G(\varepsilon_X)\circ GF(g)\circ\eta_Y \\
 &= G(\varepsilon_X)\circ\eta_{GX}\circ g \\
 &= 1_{GX}\circ g = g\end{align}</math>
:hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ<sup>−1</sup> = Ψ.

===Hom-set adjunction induces all of the above===

Given functors ''F'' : ''D'' → ''C'', ''G'' : ''C'' → ''D'', and a hom-set adjunction Φ : hom<sub>''C''</sub>(''F''&minus;,&minus;) → hom<sub>''D''</sub>(&minus;,''G''&minus;), one can construct a counit–unit adjunction

:<math>(\varepsilon,\eta):F\dashv G</math>&nbsp;,

which defines families of initial and terminal morphisms, in the following steps:

*Let <math>\varepsilon_X=\Phi_{GX,X}^{-1}(1_{GX})\in\mathrm{hom}_C(FGX,X)</math> for each ''X'' in ''C'', where <math>1_{GX}\in\mathrm{hom}_D(GX,GX)</math> is the identity morphism.
*Let <math>\eta_Y=\Phi_{Y,FY}(1_{FY})\in\mathrm{hom}_D(Y,GFY)</math> for each ''Y'' in ''D'', where <math>1_{FY}\in\mathrm{hom}_C(FY,FY)</math> is the identity morphism.
*The bijectivity and naturality of Φ imply that each (''GX'', ε<sub>''X''</sub>) is a terminal morphism from ''F'' to ''X'' in ''C'', and each (''FY'', ''η''<sub>''Y''</sub>) is an initial morphism from ''Y'' to ''G'' in ''D''.
*The naturality of Φ implies the naturality of ε and ''η'', and the two formulas
:<math>\begin{align}\Phi_{Y,X}(f) = G(f)\circ \eta_Y\\
\Phi_{Y,X}^{-1}(g) = \varepsilon_X\circ F(g)\end{align}</math>
:for each {{itco|''f''}}: ''FY'' → ''X'' and {{itco|''g''}}: ''Y'' → ''GX'' (which completely determine Φ).

*Substituting ''FY'' for ''X'' and ''η''<sub>''Y''</sub> = Φ<sub>''Y'', ''FY''</sub>(1<sub>''FY''</sub>) for ''g'' in the second formula gives the first counit–unit equation
:<math>1_{FY} = \varepsilon_{FY}\circ F(\eta_Y)</math>,
:and substituting ''GX'' for ''Y'' and ε<sub>X</sub> = Φ<sup>−1</sup><sub>''GX, X''</sub>(1<sub>''GX''</sub>) for ''f'' in the first formula gives the second counit–unit equation
:<math>1_{GX} = G(\varepsilon_X)\circ\eta_{GX}</math>.