Let $G$ be a Lie group acting on two smooth manifolds $M$ and $N$ such that both $M/G$ and $N/G$ are smooth manifolds. Suppose that $f: M\to N$ is a smooth proper $G$-map. Then it induces a smooth map $\bar f: M/G\to N/G$. My question is
Is the induced map $\bar f$ proper?
Here, proper means preimages of compact subsets are compact. All manifolds are assumed to be Hausdorff.
Assuming the group actions to be smooth, the quotient maps $p:M\to G\backslash M$ and $q:N\to G\backslash N$ are open. Let $C\subset G\backslash N$ be compact. For $q(n)\in C$, $n\in N$, pick a relatively compact open neighborhood $U_n$ of $n$ in $N$. The sets $q(U_n)$, $q(n)\in C$, form an open covering of $C$, so finitely many $U_{n_1},\dots,U_{n_k}$ suffice. Then $K=\overline{U_{n_1}}\cup\dots\cup \overline{U_{n_k}}$ is a compact set in $N$ such that $C\subset q(K)$. The preimage $f^{-1}(K)$ is compact in $M$ and its image $p(f^{-1}(K))$ in $G\backslash M$, which is compact, contains the preimage of $C$, which is closed, hence compact.
