  We give an explicit formula, as a formal differential operator, for quantum
microformal morphisms of (super)manifolds that we introduced earlier. Such
quantum microformal morphisms are essentially oscillatory integral operators or
Fourier integral operators of a particular kind. They act on oscillatory wave
functions, whose algebra extends the algebra of formal power series in Planck's
constant.
  In the classical limit, quantum microformal morphisms reproduce, as the main
term of the asymptotic, the nonlinear pullbacks of functions with respect to
`classical' microformal morphisms. We found these nonlinear transformations of
functions in search of $L_{\infty}$-morphisms for homotopy Poisson structures.
