  Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a
2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i)
$\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii)
$\mathfrak{T}_\bullet$ preserves proneness of morphisms in
$\mathcal{C}^\downarrow$. Then $\mathfrak{T}_\bullet$ preserves fibrations and
opfibrations in $\mathcal{C}$.
  The proof takes Street's characterization of (e.g.) opfibrations as
pseudoalgebras for 2-monads $\mathfrak{L}_B$ on slice categories
$\mathcal{C}/B$ and develops it by defining a 2-monad $\mathfrak{L}_\bullet$ on
$\mathcal{C}^\downarrow$ that takes change of base into account, and uses known
results on the lifting of 2-functors to pseudoalgebras.
