Editing Adjoint functors (section)

===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>.