We provide a categorical presentation of the Fusion calculus. First, 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 aa 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 Fusioncalculus which is both compositional and fully abstract: two processes have the same semantics iff they are bisimilar, that is, hyperequivalent.