Chapter 4: The Self-Enrichment Functor

A category enriches over itself when its morphism spaces are themselves objects of the category. Book II, Part VIII proved that Category τ has exactly this property: Hom(A,B) ∈ Obj(τ) for every pair of objects A, B, and the morphism spaces carry the same bipolar decomposition as the objects they connect. This chapter receives that result, extracts its structural content, and defines the enrichment functor F_E that iterates self-enrichment to produce new layers. Each application of F_E creates a category whose Hom objects live one enrichment level higher. The iteration E₀ → E₁ → E₂ → E₃ is the architectural spine of the entire series.