Tangent functor on microformal morphisms

We show how the tangent functor extends naturally from ordinary smooth maps to "microformal" (or "thick") morphisms of supermanifolds, a notion that we introduced earlier. Microformal morphisms generalize ordinary maps and they can be seen as formal canonical relations between the cotangent bundles. They are specified by generating functions depending as arguments on coordinates on the source manifold and momentum variables on the target manifold, and which are formal power expansions in momenta. Microformal morphisms act on functions by pullbacks that are non-linear transformations. (The initial motivation that led us to introducing such a notion was constructing L∞-morphisms of higher Koszul brackets.) Constructions obtained in this paper give, in particular, non-linear pullbacks of differential forms.