Functor whose essential image is a cosieve? Definitions
An object $d \in Obj(\mathcal D)$ is in the essential image of $F$ if there exists some $c \in Obj(\mathcal C)$ such that $d \cong F c$.
A sieve in $\mathcal D$ is a full subcategory of $\mathcal D$ such that if $y$ is in the sieve and there is a morphism $\varphi : x \to y$, then $x$ is in the sieve.
Since sieves are required to be full, they can be regarded as subsets of $Obj(\mathcal{D})$.
A cosieve is the dual: if $x$ is in the cosieve and there is a morphism $\varphi : x \to y$, then $y$ is in the cosieve.
Question
Is there a word for a functor $F : \mathcal C \to \mathcal D$ satisfying the following three equivalent conditions:

*

*the objects in the essential image of $F$ constitute a cosieve in $\mathcal D$,


*the objects outside the essential image of $F$ constitute a sieve in $\mathcal D$,


*if there is a morphism $Fc \to d$, then $d$ is in the essential image of $F$.
Of course a word for the dual is also fine.
 A: *

*A discrete opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d'$ is the image of a unique morphism $c \to c'$ on the nose (we need $Fc' = d'$).


*A Grothendieck opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d'$ is the image of a universal ("cartesian") morphism $c \to c'$ on the nose (we need $Fc' = d'$).
Thus, every discrete opfibration is a Grothendieck opfibration.
If the opfibration is cloven, meaning that a choice of cartesian morphisms has been made, then for every morphism $\varphi : d \to d'$ in $\mathcal D$, we get a functor $E^{-1}(\varphi) : E^{-1}(d) \to E^{-1}(d')$. Universality of the lifting of the arrow is what makes $E^{-1}$ a pseudofunctor.


*A Street/weak opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d$ is the image of a universal morphism $c \to c'$ up to isomorphism (we need $Fc' \cong d'$).
Thus, every Grothendieck opfibration is a Street opfibration.


*A functor $F : \mathcal C \to \mathcal D$ as described in the question, has the property that every morphism $Fc \to d$ is the image of some morphism $c \to c'$ up to isomorphism (we need $Fc' \cong d'$). Lacking universality, even if this were cloven (in the sense that a choice of liftings of morphisms to $\mathcal C$ has been made), we cannot expect $E^{-1}$ to satisfy the functor laws up to isomorphism.
Edit: The question did not require the morphism to be in the image of $F$. Of course this will be automatic if $F$ is full.
So I guess a reasonable name (when morphisms are also lifted) would be weak ad hoc opfibration:

*

*Weak because it lifts arrows up to isomorphism,

*Ad hoc because we can choose liftings for arrows in an ad hoc manner, not caring about the bigger picture, since we need not satisfy pseudofunctoriality.

