  In 2014, Voronov introduced the notion of thick morphisms of (super)manifolds
as a tool for constructing $L_{\infty}$-morphisms of homotopy Poisson algebras.
Thick morphisms generalise ordinary smooth maps, but are not maps themselves.
Nevertheless, they induce pull-backs on $C^{\infty}$ functions. These
pull-backs are in general non-linear maps between the algebras of functions
which are so-called 'non-linear homomorphisms'. By definition, this means that
their differentials are algebra homomorphisms in the usual sense. The following
conjecture was formulated: an arbitrary non-linear homomorphism of algebras of
smooth functions is generated by some thick morphism. We prove here this
conjecture in the class of formal functionals. In this way, we extend the
well-known result for smooth maps of manifolds and algebra homomorphisms of
$C^{\infty}$ functions and, more generally, provide an analog of classical
'functional-algebraic duality' in the non-linear setting.
