The configuration spaces Conf(X,n) of a topological space X play an important role in geometric topology. Recent developments have shown that, for fixed spaces X and Y, it can be fruitful to study compatible families of maps from Conf(X,n) to Conf(Y,n) as n varies. There are several ways of assembling together configuration spaces and maps thereof as n varies, including embedding calculus, configuration categories, and Ran spaces. In this talk we show that the mapping spaces arising from the latter two approaches agree. In particular, we prove that the natural map Aut_{/Fin}(Conf(R^d))-->Aut_{/Fin}(Entr(Ran(R^d))) is an equivalence. Here Entr(Ran(R^d)) is the opposite of the exit-path ?-category of the Ran space of R^d equipped with its usual stratification, and Conf(R^d) is the configuration category of R^d. As an application, we prove a corrected version of a conjecture of Ayala, Francis and Tanaka concerning the relationship between Top(d) and Aut_{Fin}(Entr(Ran(R^d))). The main ingredient is a stratified nerve theorem, establishing a presentation of the enter path ?-category of a conically stratified space X-->P with locally weakly contractible strata as a localization of the poset associated with a sufficiently good open cover of X. As a by-product of these methods, we also obtain a much simpler proof of the exodromy equivalence for conically stratified spaces with weakly contractible strata. This is a joint work with Alexander Kupers.