Two-variable affine extension before hypercompletion

For an $F$-algebra $A$, let $I_A$ be the category whose objects are smooth finitely presented $F$-algebras $B$ with a map $B\to A$, and whose arrows commute with that map. It is nonempty since $F\to A$ is an object. For $(B\to A,C\to A)$, the tensor product $B{\otimes }_{F}C$, mapped to $A$ by $b\otimes c\mapsto f(b)g(c)$, is a coproduct in $I_A$ by [F3]. It is still smooth finitely presented: its spectrum is the affine product, whose projection is smooth by base change and whose structure map is smooth by composition; the tensor of two finite presentations is a finite presentation. The comma category from $(B,C)$ to the diagonal $I_A\to I_A\times I_A$ has the initial object given by the two tensor inclusions. Its nerve is contractible. Thus [F4] makes this diagonal cofinal and $I_A$ sifted. By the pointwise formula and finite-product commutation in [F4], the counit for the product is the composite equivalence $L(j^{\ast }P\times j^{\ast }P)(A)\simeq ({\mathrm{colim}}_{I_A}P(B))\times ({\mathrm{colim}}_{I_A}P(B))\simeq P(A)\times P(A)$, the last equivalence being [F1]. These are the natural counits, so the equivalence is natural in $A$. The Kan adjunction now gives exactly the displayed equivalence of mapping spaces into $Y_i$. It transports paths between the specified endpoints, not merely their objectwise component equalities.

Apply step 1.1 to the two given affine operations and their smooth-affine homotopy. Extend that one path using the same affine-basis right Kan extension and source unit as [F2]. Right Kan extension is computed by limits, which preserve products: the limit of a diagram of pairs is a pair of compatible cones, equivalently the product of the two limits. Ordinary sheafification also preserves finite products by [F4]; hence its source unit on $P\times P$ is the product of the two units on $P$. The target is the original sheaf $Y_i$ by [F2]. Consequently the extended path has exactly the claimed endpoints on the ordinary sheafified source and, after its unit, on every global perfect K-space. Extension and restriction are functors, so they preserve composition and the supplied natural sums, source addition and coefficient cup products in those endpoints. This argument applies to the corresponding extended operations specified in the Statement; it does not assert that arbitrary nonaffine operations are determined merely by equality of affine components. The affine-basis formula in [F2] includes nonaffine intersections on nonseparated qcqs schemes. Thus the conclusion holds in the full stated scope before hypercompletion.