  Functor lifting along a fibration is used for several different purposes in
computer science. In the theory of coalgebras, it is used to define coinductive
predicates, such as simulation preorder and bisimilarity. Codensity lifting is
a scheme to obtain a functor lifting along a fibration. It generalizes a few
previous lifting schemes including the Kantorovich lifting. In this paper, we
seek a property of functor lifting called fiberedness. Hinted by a known result
for Kantorovich lifting, we identify a sufficient condition for a codensity
lifting to be fibered. We see that this condition applies to many examples that
have been studied. As an application, we derive some results on
bisimilarity-like notions.
