Double-adjoint comparison transformation
= Double-adjoint comparison transformation
{title2=$\theta$}
For $F\dashv G\dashv H$ with full and faithful $G$, write $\eta,\varepsilon$ for the first unit and counit and $\alpha,\beta$ for the second. The two composites
$$
H\xrightarrow{H\eta}HGF\xrightarrow{(\alpha F)^{-1}}F,
\qquad
H\xrightarrow{(\varepsilon H)^{-1}}FGH\xrightarrow{F\beta}F
$$
coincide. Their common value is the double-adjoint comparison transformation $H\to F$.