This paper provides both a fully abstract (domain-theoretic) model for the [pi]-calculus and a universal (set-theoretic) model for the finite [pi]-calculus with respect to strong late bisimulation and congruence. This is done by considering categorical models, defining a metalanguage for these models, and translating the [pi]-calculus into the metalanguage. A technical novelty of our approach is an abstract proof of full abstraction: The result on full abstraction for the finite [pi]-calculus in the set-theoretic model is axiomatically extended to the whole [pi]-calculus with respect to the domain-theoretic interpretation. In this proof, a central role is played by the description of nondeterminism as a free construction and by the equational theory of the metalanguage.