Fully faithful adjoint criterion

For $F⊣G$, the right adjoint $G$ is full and faithful exactly when the counit $FG\to 1$ is an isomorphism. Dually, $F$ is full and faithful exactly when the unit $1\to GF$ is an isomorphism.

For $F⊣G⊣H$ with full and faithful $G$, write $\eta ,\epsilon$ for the first unit and counit and $\alpha ,\beta$ for the second. The two compositescoincide. Their common value is the double-adjoint comparison transformation $H\to F$.