Geometric constructions preserve fibrations

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) $\mathsf{c}\mathsf{o}\mathsf{d}{\mathfrak{T}}_{\bullet }=\mathsf{c}\mathsf{o}\mathsf{d}$ 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 $\mathcal{C}\mathrm{/}B$ 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.