The "nonlinear pullback" of functions and a formal category extending the category of supermanifolds

We introduce mappings between function spaces on smooth (super)manifolds, which are generally nonlinear and which generalize the pullbacks with respect to smooth maps. The construction uses canonical relations and generating functions. The formal category arising here is a "formal thickening" of the usual category of supermanifolds. (It is very close to the category of symplectic micromanifolds and their micromorphisms considered recently by A. Weinstein and A. Cattaneo -- B. Dherin -- A. Weinstein.) We also give a parallel construction for odd functions, based on odd symplectic geometry. As an application, we show that such a mapping gives an L∞-morphism of the algebras of functions on homotopy Schouten manifolds if the master Hamiltonians are related by a canonical relation. There is a parallel statement for homotopy Poisson manifolds. We discuss possible applications to L∞-bialgebroids.