 Discrete opfibration

A functor with a unique lift, from each specified object $c$, of every arrow $Fc\to d$. The lifted arrow's target is part of the uniqueness requirement. This is dual to a discrete fibration. Lifting a cocone of a finite connected category gives a cocone upstairs: lifts of its legs have a common target because uniqueness identifies targets along every diagram arrow. Thus a discrete opfibration over a weakly filtered category has weakly filtered domain, including the possibility of an empty domain.