I have a 2-functor $F : \mathcal{C} \to \mathcal{D}$ which is "lax full" in the sense that for every 1-cell $a : FX \to FY$ of $\mathcal{D}$, there is a 1-cell $\overline{a}$ in $\mathcal{C}$ and a 2-cell $F\overline{a} \to a$ in $\mathcal{D}$, natural in $a$. The essential image of $F$ between $FX$ and $FY$ are precisely those $a$ for which $F\overline{a} \to a$ is an isomorphism.
I'm wondering if this notion has ever appeared in the literature, or whether it has an established name. References would be much appreciated!
EDIT: Actually, the situation I have is that the action $F : \mathcal{C}(-,-) \to \mathcal{D}(F-, F-)$ has a right adjoint $\overline{-} : \mathcal{D}(F-, F-) \to \mathcal{C}(-,-)$, and the counit is the natural transformation $F\overline{a} \to a$ mentioned above.