Chapter 59: Functors and Natural Transformations

the relevant chapter earned the category Cat_τ (the relevant definition, I.D51): a thin, countable category whose objects are τ-indices and whose morphisms are τ-arrows — equivalence classes of τ-holomorphic functions identified by the τ-Identity Theorem (the relevant theorem, I.T21). This chapter develops the next layer: τ-functors (the relevant definition, I.D52), natural transformations (the relevant definition, I.D53), and the Yoneda embedding (the relevant definition, I.D54). The Yoneda Lemma (the relevant theorem, I.T23) shows Nat(y(X), F) ≅ F(X) for every presheaf F and object X. In the thin setting, representable presheaves are subterminal and the lemma takes a clean form, yet retains its full force: objects of Cat_τ are completely determined by their network of relationships to all other objects. These constructions set the stage for limits and sites

and the earned topos .