  We introduce mappings between spaces of functions on (super)manifolds that
generalize pullbacks with respect to smooth maps but are, in general, nonlinear
(actually, formal). The construction is based on canonical relations and
generating functions. (The underlying structure is a formal category, which is
a "thickening" of the usual category of supermanifolds; it is close to the
category of symplectic micromanifolds and their micromorphisms considered
recently by A. Weinstein and A. Cattaneo--B. Dherin--Weinstein.) There are two
parallel settings, for even and odd functions. As an application, we show how
such nonlinear pullbacks give $L_{\infty}$-morphisms for algebras of functions
on homotopy Schouten or homotopy Poisson manifolds.
