The completeness theorem implies that basic notions from Hilbert spaces carry over to any dagger compact category. The typical language employed, however, changes. The notion of a basis is given in terms of a coalgebra. Given an object "A" from a dagger compact category, a basis is a comonoid object formula_12. The two operations are a "copying" or comultiplication δ: "A" → "A" ⊗ "A" morphism that is cocommutative and coassociative, and a "deleting" operation or counit morphism ε: "A" → "I" . Together, these obey five axioms: