Dual formulation of the lemma on substitution homomorphisms

This paper contains a nonstandard formulation of the well-known lemma on substitution homomorphisms stated as the canonical duality between the family of all smooth mappings of one smooth manifold into another and the family of all homomorphisms of algebras of smooth scalar functions on these manifolds. This formulation gives the lemma the maximum possible generality and emphasizes the fundamental symmetry of the problem: the duality between “conjugation” (transition from mappings of manifolds to homomorphisms of algebras of smooth functions on them) and “co-conjugation” (transition from homomorphisms to mappings).