Let \({\mathcal Mf}^ +_ m\) be the category of oriented \(m\)-dimensional manifolds and orientation preserving local diffeomorphisms. Consider two natural bundles \(F\) and \(G\) on \({\mathcal Mf}^ +_ m\) and a system \(A_ M: C^ \infty FM\to C^ \infty GM\) for all \(M\in \hbox{Ob}{\mathcal Mf}^ +_ m\). Such a system \(A\) is called a natural operator, if it commutes with the morphisms from \({\mathcal Mf}^ +_ m\), while \(A\) is said to be infinitesimally natural, if the Lie derivative \(L_ XA_ M\) vanishes for every vector field \(X\) on each \(M\in \hbox{Ob}{\mathcal Mf}^ +_ m\). The main result of the paper is that every infinitesimally natural operator is natural. The proof is heavily based on the nonlinear Peetre theorem established by the second author [Ann. Global Anal. Geom. 6, No.3, 273-283 (1988; Zbl 0636.58042)]. In conclusion, the authors deduce a similar result for the gauge-natural operator in the sense of D. J. Eck [Mem. Am. Math. Soc. 247, 48 p. (1981; Zbl 0493.53052)].