Fullness and faithfulness of a right adjoint are detected by its counit

If $G$ is faithful and $r{\epsilon }_d=s{\epsilon }_d$, then $G(r)G({\epsilon }_d)=G(s)G({\epsilon }_d)$; composing with ${\eta }_{Gd}$ and using the triangle identity gives $G(r)=G(s)$, hence $r=s$. Thus each ${\epsilon }_d$ is epic.

Conversely, if every ${\epsilon }_d$ is epic and $G(r)=G(s)$, naturality gives $r{\epsilon }_d={\epsilon }_{e}FG(r)={\epsilon }_{e}FG(s)=s{\epsilon }_d$, so $r=s$. Hence $G$ is faithful.

If $G$ is full, lift ${\eta }_{Gd}:Gd\to GFGd$ to $q_d:d\to FGd$ with $G(q_d)={\eta }_{Gd}$. Naturality of $\epsilon$ and the first triangle identity give $q_d{\epsilon }_d={\epsilon }_{FGd}FG(q_d)={\epsilon }_{FGd}F({\eta }_{Gd})=1_{FGd}$, so ${\epsilon }_d$ is split monic.

Conversely, choose $q_d$ with $q_d{\epsilon }_d=1_{FGd}$. Then $G(q_d)$ is the inverse of $G({\epsilon }_d)$, whose right inverse is ${\eta }_{Gd}$ by the triangle identity, so $G(q_d)={\eta }_{Gd}$. For $h:Gd\to Ge$, the morphism $r:={\epsilon }_{e}F(h)q_d:d\to e$ satisfies $G(r)=G({\epsilon }_e)GF(h){\eta }_{Gd}=h$, using naturality of $\eta$ and the triangle identity. Thus $G$ is full.

A fully faithful $G$ makes ${\epsilon }_d$ both epic and split monic by steps 1.1 and 1.3; if $q_d{\epsilon }_d=1$, epicity gives ${\epsilon }_d{q}_d=1$, so ${\epsilon }_d$ is an isomorphism. Conversely, an invertible counit is epic and split monic, so steps 1.2 and 2.1 make $G$ fully faithful.

Passing to opposite categories exchanges the counit with the unit, faithful with faithful, epic with monic, and split monic with split epic, proving the dual assertions.