The following statement may be "well known but not well known enough", and my question is which reference would state it explicitly:
The construction of Jet bundles is a comonad on suitable bundles over a given base $X$. A differential operator $D \colon \Gamma_X(E_1) \to \Gamma_X(E_2)$ is equivalently a morphism of bundles of the form $\tilde D \colon Jet(E_1) \to E_2$ (the associated differential operator is $\phi \mapsto \tilde D\circ j_\infty(\phi)$). Under this identification, the composition $D_2\circ D_1$of two differential operators is given by the coKleisli composite, namely by the composite
$$ Jet(E_1) \to Jet(Jet(E_1)) \stackrel{Jet(\tilde D_1)}{\longrightarrow} Jet(E_2) \stackrel{\tilde D_2}{\longrightarrow} E_3 $$
where the first morphism is the coproduct of the Jet comonad.
In the context of differential geometry, the article
- Joseph Krasil'shchik, Alexander Verbovetsky, Homological Methods in Equations of Mathematical Physics (arXiv:math/9808130)
has essentially all the ingredients for this statement (p. 13,14,17), but does not make it explicit in the above form. In the algebraic context there may be more references that state it this way or almost state it this way (the statement that $Jet$ is a comonad for sure, but how about the differential operators being coKleisli maps?). Whatever references you are aware of, please let me know.