Categorical H: Histories as Objects, Erasure as Functor

This paper categorifies Tseglin 2026a/b/c/d/e. In 2026a, we proved no unital map T → E for H = S^1. In 2026b, we generalized to the triple H^n(T(H)) = (dim H^n, dim C^n, 1) for all finite H, all n. In 2026c, we defined the Bridge Conjecture. In 2026d, we proved the eraser must be CPTP decoherence. In 2026e, we proved all four follow from one Unifier Axiom: [x,y] = 1, and no unital φ: T(H) → E preserves [x,y]. Here we lift H from a space to a category. Histories become objects. Maps between histories become morphisms. T becomes a functor. Erasure becomes a functor that cannot be unital. We prove the No-Go is natural: it holds for every object, and it commutes with every morphism. We prove the triple is a functorial invariant, and the Unifier is a natural transformation that never vanishes. This does not add a new experiment. It adds the language in which the next papers, char p and non-finite H, can be stated. Tseglin 2026f = Ch. 141. Cites Tseglin 2026a DOI 10.5281/zenodo.23977165, 2026b DOI 10.5281/zenodo.23980498, 2026c DOI 10.5281/zenodo.23980738, 2026d DOI 10.5281/zenodo.23981860, 2026e DOI 10.5281/zenodo.23982063.