Equivalence between twisted bimodules and loosely discrete opfibrations

Let $\lModl$ be the 2-category of loosely discrete opfibrations and $\TwModl$ the 2-category of twisted bimodules, with the 2-functor $\twsection{-}: \lModl \to \TwModl$ given by the sections construction. For a twisted bimodule $M: \dbl{D} \twbimodto \dbl{E}$, the collage construction produces a loosely discrete opfibration whose source and target are respectively equivalent to $\dbl{D}$ and $\dbl{E}$. Sections–collage equivalence conjecture. The 2-functor

$$\twsection{-}: \lModl \to \TwModl$$

is an equivalence of 2-categories, with inverse given by the collage construction. The expected inverse relationship is straightforward on the underlying objects and elements, while recovering the comparison cells requires a coherence principle for twisted bimodules; the correspondence of the remaining data is likewise expected modulo these coherence difficulties.