In the paper under review the author proves the existence of a Quillen equivalence between the covariant model structure of the slice category \(\mathbf{S}/B\) of simplicial sets over a fixed simplicial set \(B\) and a certain localization on the category of simplicial presheaves on the simplex category \(\Delta /B\). Extending this result he also shows a new Quillen equivalence between such a covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category \(\mathfrak{C}[B].\) Here, the functor \(\mathfrak{C}[-]\), from simplicial sets to simplicial categories, stands for the left adjoint of \(N_{\Delta}\), the simplicial or homotopy coherent nerve functor, see J. Lurie’s work in [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)], or J.-M. Cordier’s paper [Cah. Topologie Géom. Différ. Catégoriques 23, 93–112 (1982; Zbl 0493.55009)]. Besides establishing some nice results on localizations of simplicial categories and quasi-categories the author also analyzes the relationship with the so called Lurie’s straightening theorem.