Unit and counit of an adjunction ($\eta ,\epsilon$)

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

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.