In general, the counit of an adjunction is an isomorphism if and only if the right-adjoint is fully faithful (dually the unit is an iso iff the left-adjoint is fully-faithful). So, your question is easily seen to be equivalent to asking "When is $F^{*}$ fully-faithful? In topos-theory lingo, when is the induced geometric morphism $mathbf{F}:Set^{C^{op}} to Set^{D^{op}}$ satisfies $F^*$ is faithful, then $mathbf{F}$ is said to be a SURJECTION of topoi. In this setting, this is equivalent to every object in $D$ being a retract of an object of the form $F(C)$.
Ok, so how about asking for $F^*$ to also be full? $F^*$ being faithful AND full means you are looking at what is called a CONNECTED geometric morphism of topoi. What properties $F$ do we need to ensure this? This is in general a hard problem. However, there are at least sufficient conditions. Given $F$, you first construct the category $Ext_{F}$ of "F-extracts"- these are quadruples $(U,V,r,i)$ with $U in C$, $V in D$, $r:FU to B$, and $i:V to FU$ such that $ri=1$, with the evident morphisms. There is a canonical functor $tilde F:Ext_{F} to D$ which sends $(U,V,r,i) mapsto V$. Denote by $Ext_F(V)$ the fiber over $V$ of this functor. Then if $tilde F$ is full and each $Ext_F(V)$ is a connected category, then $mathbf{F}$ is a connected morphism.
This is in "Sketches of an Elephant" C.3.3.
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