  We show that, for any simplicial space $X$, the $\infty$-category of culf
maps over $X$ is equivalent to the $\infty$-category of right fibrations over
$\operatorname{sd}(X)$, the edgewise subdivision of $X$ (when $X$ is a Rezk
complete Segal space or 2-Segal space, this is the twisted arrow category of
$X$). We give two proofs of independent interest; one exploiting comprehensive
factorization and the natural transformation from the edgewise subdivision to
the nerve of the category of elements, and another exploiting a new
factorization system of ambifinal and culf maps, together with the right
adjoint to edgewise subdivision. Using this main theorem, we show that the
$\infty$-category of decomposition spaces and culf maps is locally an
$\infty$-topos.
