Let $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{A}$ be a pair of adjoint functors on some categories.
The following is well-known:
The right adjoint functor $G$ is fully faithful if and only if the counit $\varepsilon_Y: FG(Y) \to Y$ is an isomorphism for any object $Y$ from $\mathcal{B}$. In this case, the left adjoint functor $F$ is dense (or essentially surjective).
Question 1: Are there some nice conditions under which the converse true? Assuming the left adjoint functor $F$ to be dense, what else do we need to ensure that the right adjoint functor $G$ is fully faithful?
This is a coarse version of the next question. For the next one, let us fix an object $Y$ from $\mathcal{B}$. If the counit $\varepsilon_Y$ is an isomorphism, then $Y$ lies in the essential image of $F$.
Question 2: Are there some nice conditions such that the converse true? In other words, when is it true that $\varepsilon_{F(X)}: FGF(X) \to F(X)$ is an isomorphism?
An example of such a condition is that $F$ is full. It is known that an equivalent condition is that the unit $\eta_X: X \to GF(X)$ is a split epimorphism (that is, it admits a right inverse) for any $X$ from $\mathcal{A}$. By the counit-unit-equation $\varepsilon_{F(X)} \circ F(\eta_X) = id_{F(X)}$, the unit $F(\eta_X)$ is a split monomorphism and $\varepsilon_{F(X)}$ a split epimorphism. It then follows that $F(\eta_X)$, and thus $\varepsilon_{F(X)}$, are both isomorphisms.
I would like to know whether there are other conditions than the fullness of $F$ or even some categorical characterization for the essential image of $F$ to be precisely given by the objects $Y$ from $\mathcal{B}$ such that $\varepsilon_Y$ is an isomorphism.
A positive answer to Question 2 would imply a positive answer to Question 1. In particular, if the left adjoint functor $F$ is full and dense, then $G$ must be full and faithful.
A reference to the 'well-known' properties of adjoint functors is Proposition 2.4 in ncatlab.