Smooth and continuous real singular cohomology agree

For an ordered open cover $M=U\cup V$, restriction to smooth simplices gives a diagram from the ordinary short exact small-dual row to the smooth row. Every arrow commutes: restrictions and the $V-U$ difference evaluate the same cochain on the same smooth simplex, and signed coboundaries use the same faces. If an ordinary overlap cocycle $c$ is lifted to $e$ with $\delta e=a(d)$, its smooth restriction is a lift of the smooth restriction of $c$, and its differential is the smooth restriction of $a(d)$. Thus the two small-dual lift-differential connectors commute. The squares with actual small-chain inclusions also commute on every smooth small simplex. Their cohomology maps are isomorphisms by [F1], so transporting connectors through their inverses preserves the square. All arrows in the two Mayer–Vietoris sequences therefore commute with restriction.

By [F4] the countable product interface and its comparison square hold under [A1]. The empty manifold has zero chain complexes and zero cohomology on both sides, so its comparison is an isomorphism. Every rational open box and rational half-box is convex; [F3] gives the local comparison isomorphisms, including the zero-dimensional point. Both functors are invariant under diffeomorphisms by functoriality, since the pullbacks of inverse maps are inverse.

All hypotheses of [F5] are now supplied by steps 1.1 and 1.2. Apply its boundaryless version to obtain the claimed isomorphism there and its half-box version to obtain it for manifolds with boundary. Naturality is the already defined square [F2], without any choices of smoothing or inverses on cochains.

Negative degrees have zero groups; in degree zero the same initial Mayer–Vietoris segments apply. Empty opens and overlaps were included in [F1]. On a point the restriction complex map is identity. Degenerate simplices are retained in both complexes and their common face equations. Countable choice is used only through the product and exhaustion/globalization arguments; no arbitrary vector-space dual exactness, full AC, or prescribed-face smoothing into a boundary target is used.