Let us have two symplectic manifolds $(M, \,\omega)$ and $(N, \,\omega')$ and morphism between them: $$ \varphi \ :\ M \to N.$$ Then we geometrically quantize these systems:

we add a prequantum line bundle, $ L_M \to M$ and $L_N \to N$ correspondingly;

choose some real polarizations $P_M$ and $P_N$ such that $(\varphi_*P_M)(x) \neq P_N(x), \, \text{for }x \in \varphi(M)$;

let the quotient maps $\pi_{M}\ :\ M \to M/P_M$ and $\pi_N\ :\ N \to N/P_N$ be fibration.

Then the space of compactly supported sections of $L_M/P_M \otimes |T(M/P_M)|^{1/2} $ after completion realise Hilbert space $H_M$. The same procedure for $N$ gives $H_N$.

I don't understand how to build morphism between $H_M$ and $H_N$ from $\varphi$. Is there any natural way to do it?

I'm not sure whether it's a trivial question or not. I don't have deep understanding of geometric quantization, I've only read this review by Lerman. So if some comprehensible text about the question exists, I'll gladly accept a reference.

P.S. I'm reading this paper by Nekrasov which describes morphism between classical Calogero-Moser and Calogero-Sutherland models, and derives from it morphism between their quantum versions. Nekrasov briefly describes how he gets quantum morphism from classical one in the beginning of 4th section. But I don't understand it. So I'm interested whether there is a general way to do it. If it is possible only in this particular case, I'd like to get a more thoroughly explanation of Nekrasov's procedure.