Chapter 44: Yoneda Embedding as Theorem

In classical category theory, the Yoneda lemma holds for any locally small category — it is purely formal, requiring nothing about the objects or morphisms beyond their existence. In Category τ, the Yoneda embedding y : τ ↪ [τ^op, τ] is a theorem (II.T36), not an abstract generality. Three features distinguish the τ-Yoneda from its classical cousin: (i) the target category [τ^op, τ] lives inside τ itself (self-enrichment, II.D53), not in an external universe of sets; (ii) the embedding preserves the bipolar structure (II.P11); (iii) fullness is witnessed by the Code/Decode bijection (II.T35), which provides computable, finite-stage certificates. The proof rests on a single observation: probe naturality — the condition that characterized holomorphy in Part II (Remark [rem:probe-naturality], II.R12) — is exactly the Yoneda condition (Lemma [lem:probe-yoneda], II.L11). Holomorphy, probe naturality, and Yoneda representability are three names for the same structural phenomenon. This is the deep reason that holomorphy is primitive in Category τ.