An $(\mathrm{\infty },n)$-categorical straightening-unstraightening construction

We provide an $(\mathrm{\infty },n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\mathrm{\infty },n)$-category of double $(\mathrm{\infty },n-1)$-right fibrations over an $(\mathrm{\infty },n)$-category $\mathcal{C}$ and that of the $(\mathrm{\infty },n)$-functors from $\mathcal{C}$ valued in $(\mathrm{\infty },n-1)$-categories. We realize this in the form of a Quillen equivalence between appropriate model structures; on the one hand, a model structure for double $(\mathrm{\infty },n-1)$-right fibrations over a generic precategory object $W$ in $(\mathrm{\infty },n-1)$-categories and, on the other hand, a model structure for $(\mathrm{\infty },n)$-functors from its homotopy coherent categorification $\mathfrak{C}W$ valued in $(\mathrm{\infty },n-1)$-categories.