We provide a categorical presentation of the Fusion calculus. First we give Working in a suitable category of presheaves, we describe the syntax as initial algebra of a signature endofunctor, and the semantics as coalgebras of a ‘behaviour’ endofunctor. To this end, we first give a a new, congruence-free presentation of the Fusion calculus; then, the behaviour endofunctor is constructed by adding in a systematic way a notion of ‘state’ to the intuitive endofunctor induced by the LTS. Coalgebras can be given a concrete presentation as ‘stateful indexed labelled transition systems’; the bisimilarity over these systems is a congruence, and corresponds to hyperequivalence. Then, we model the labelled transition system of Fusion by abstract categorical rules. As a consequence, we get a semantics for the Fusion calculus which is both compositional and fully abstract: two processes have the same semantics iff they are bisimilar, that is, hyperequivalent. © 2008 Elsevier B.V. All rights reserved.