Editing Adjoint functors (section)

===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.