Chapter 62: Cartesian Product as Earned Bi-Functor

the relevant chapter earned the topos E_τ = PSh(Cat_τ) (the relevant definition, I.D59), and the relevant chapter constructed finite limits in Cat_τ (the relevant definition, I.D55), where binary products are multiplication X ×_τ Y := X · Y with projections via the NF address encoding (the relevant definition, I.D16). This chapter lifts products to the presheaf topos. The categorical product (the relevant definition, I.D60) of presheaves P and Q is pointwise: (P × Q)(X) = P(X) × Q(X), with pairing realized by the constructive encoding of Part V (Definition [cor:constructive-encoding], I.C01). The product universal property (the relevant theorem, I.T26) gives a unique ⟨ f, g ⟩ : R → P × Q for any morphisms f : R → P, g : R → Q. The cartesian monoidal structure (the relevant definition, I.D61) makes (E_τ, ×, 1) a symmetric monoidal category, with every ingredient earned from the seven axioms. The wedge product ∧

will supply the second monoidal structure.