By analogy to the familiar situation in homotopy theory (i.e., (unique) path lifting in covering spaces), it is natural to consider the following. Let $P:C\to D$ be a functor. Say that $P$ has (unique) morphism lifting if given any $f\colon x \to y$ in $D$ and $y'$ in $C$ with $P(y')=y$, there exists a (unique) $f'\colon x'\to y'$ with $P(f')=f$. In other words, any morphism whose codomain can be lifted, can itself be (uniquely) lifted. One can also use the domain instead of the codomain.
I'm looking for references to this notion, or any useful comment.
Such functors can be shown to exhibit behaviour much like discrete Grothendieck fibrations, but multivalued. Details can be found here.