 Redshift shell volume at the present epoch

On a present FLRW spatial slice, a shell has volume $R_0^3{S}^2(\chi )d\chi d\Omega$. The radial light-ray relation gives $d\chi /dz=c/(R_{0}H)$, yielding the displayed expression in terms of transverse comoving distance. At emission the proper volume is smaller by $(1+z{)}^3$. For a conserved comoving population its proper density is greater by the same factor, so either epoch gives the same count.