In this paper, we propose a compositional coalgebraic semantics of the pi-calculus based on a novel approach for lifting calculi with structural axioms to coalgebraic models. We equip the transition system of the calculus with permutations, parallel composition and restriction operations, thus obtaining a bialgebra. No prefix operation is introduced, relying instead on a clause format defining the transitions of recursively defined processes. The unique morphism to the final bialgebra induces a bisimilarity relation which coincides with observational equivalence and which is a congruence with respect to the operations. The permutation algebra is enriched with a name extrusion operator delta 'a la De Brujin, that shifts any name to the successor and generates a new name in the first variable x0. As a consequence, in the axioms and in the SOS rules there is no need to refer to the support, i.e., the set of significant names, and, thus, the model turns out to be first order.