On the duality in the theory of smooth manifolds

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings $f\in \operatorname{Hom}(M,M')$ of a smooth real finite-dimensional manifold $M$ into a similar manifold $M'$ and the family of homomorphisms $\varphi$ of the algebra $C^{\infty}(M')$ of smooth scalar-valued functions on $M'$ into the analogous algebra $C^{\infty}(M)$ on $M$, $\varphi\in \operatorname{Hom}\big(C^{\infty}(M'),C^{\infty}(M)\big)$. This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form.