  We study the action of the diffeomorphism group $\Diff(M)$ on the space of
proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We
show that smooth transversal slices exist through each orbit, that the quotient
space is Hausdorff and is stratified into smooth manifolds, one for each
conjugacy class of isotropy groups.
