Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor

We prove how the universal enveloping algebra constructions for Lie–Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie–Rinehart algebra morphisms, of modules over Lie–Rinehart algebras and of the infinitesimal gauge algebra of a module.