= Redshift shell volume at the present epoch
{title2=$dV_0=cD_M^2(z)\,dz\,d\Omega/H(z)$}
On a present FLRW spatial slice, a shell has volume $R_0^3S^2(\chi)d\chi\,d\Omega$. The radial light-ray relation gives $d\chi/dz=c/(R_0H)$, 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.
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