<p>We generalise the Milnor–Moore theorem for cocommutative Hopf algebras to the locality setup developed in previous work, a setting in which vector spaces are promoted to locality vector spaces by adjoining a symmetric binary relation we call locality relation. This requires a detailed study of tensor products enhanced to the locality setting, which raise challenging questions, such as whether the locality tensor product of two locality vector spaces is a locality vector space. Related open questions arise from our constructions such as whether the quotient of locality vector spaces is a locality vector space. This we reinterpret in a group theoretic language and then in terms of short exact sequences. As in the classical case, our proof of the locality extension of the Milnor–Moore theorem uses universal properties of the (locality) tensor algebra and universal enveloping algebra. Universal properties in the classical setup generalise to the locality setup for spaces generated by a locality basis, a class which includes the Grossman–Larson Hopf algebra enhanced to the locality setup. In particular, we show that the universal enveloping algebra generated by a locality Lie algebra with a locality basis is a locality algebra and hence satisfies the expected universal property.</p>