Triangular Identities

Given Functors and , specifying an Adjunction is equivalent to specifying Natural Transformations and satisfying

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
F \arrow[r,"F\eta"] 
\arrow[dr,swap, "1_{F}"] 
& FGF \arrow[d,"\varepsilon_{F}"] \\
& F
\end{tikzcd}
\end{document}

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
G \arrow[r,"1_{G}"] 
\arrow[dr,swap, "1_{\mathcal{C}}"] 
& GFG \arrow[d,"G\varepsilon"] \\
& G
\end{tikzcd}
\end{document}

the triangular identities.

Proof

Given , let and be Unit and Counit respectively. Then corresponds under the Adjunction to , so it’s . The second identity is dual. Conversely, suppose given and satisfying triangular identities. Given we define to be and given we define to be Then is

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A\arrow[r,"\eta_{A}"] 
\arrow[dr, "f"] 
& GFA \arrow[r,"GFf"]
& GFGB \arrow[r,"G\varepsilon_{B}"]
& GB \\
& GB \arrow[ru, "\eta_{GB}"]
\arrow[rru,swap, "1_{GB}"]
\end{tikzcd}
\end{document}

so and dually And and are natural since and are.

Proposition

Suppose given an equivalence and and and Natural Isomorphisms and Then there are Natural Isomorphisms and satisfying the Triangular Identities. In particular and .

Proof

We define and take to be

Note that since

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
FGFG \arrow[r,"FG\beta"] \arrow[d,"\beta_{FG}"]
 & FG \arrow[d,"\beta"] \\
FG \arrow[r,"\beta"]
 & 1_{\mathcal{D}}
\end{tikzcd}
\end{document}

is a Commutative Diagram by naturality and is Monomorphism. Similarly The triangular identities for and are

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[column sep=huge]
F \arrow[r,"\alpha_{F}"]
\arrow[dr, swap, "1"]
& GF \arrow[r, "(\beta_{FGF})^{-1}"] 
\arrow[d,"(F_{\alpha})^{-1}"]
& FGFGF \arrow[d,"F_{\alpha_{GF}}"] \\
& F \arrow[r,"(\beta_{F})^{-1}"]
\arrow[dr, swap, "1_{F}"]
& FGF \arrow[d,"\beta_{F}"]
\\
& 
&
F
\end{tikzcd}
\end{document}

and

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[column sep=huge]
G \arrow[r,"\alpha_{G}"] 
\arrow[dr,swap,"1_{\mathcal{C}}"]
& GFG \arrow[r,"(GFG\beta)^{-1}"]
\arrow[d,"(\alpha_{G})^{-1}"]
& GFGFG \arrow[d,"(GF\alpha_{G})^{-1}=(\alpha_{GFG})^{-1}"]\\
& G \arrow[r,"(GB)^{-1}"]
\arrow[dr,swap,"1_{G}"]
& GFG \arrow[d,"G\beta"] \\
& & G
\end{tikzcd}
\end{document}

so and are the unit and counit of Adjunction and and are the unit and counit of .