Is there a commonly agreed upon terminology for a functor $F:A\to B$ such that all $F(X)$ are isomorphic in $B$, where $X$ is an object of $A$ ? I would guess something like "quasi-constant" or "essentially constant" would make sense, but no result came up when I tried to search for it.
I'm especially interested in a terminology that applies when $F$ is not isomorphic to a constant functor (for instance, if $A=B$ is the category of sets with $n$ elements, then the identity functor is "quasi-constant" in the above sense since all sets with $n$ elements are isomorphic, but it is not isomorphic as a functor to any constant endofunctor of $A$).