Product on ordinary differential cohomology

For $x\in {\widehat{H}}^p(M;\mathbb{Z})$ and $y\in {\widehat{H}}^q(M;\mathbb{Z})$ define $x⌣y:={\mathrm{\Phi }}_M^{-1}({\mathrm{\Phi }}_M(x){⌣}_{\mathrm{D}\mathrm{e}\mathrm{l}}{\mathrm{\Phi }}_M(y))$. The inverse exists by [F2], and bilinearity follows by applying the homomorphisms ${\mathrm{\Phi }}_M$, the bilinear Deligne product, and ${\mathrm{\Phi }}_M^{-1}$.

Naturality follows by applying the naturality square for $\mathrm{\Phi }$ and the pullback-natural Deligne product in [F1]. Associativity and graded commutativity follow by applying the injective map ${\mathrm{\Phi }}_M$ to the two sides and using the corresponding Deligne identities. Since [F2] commutes with $I$, $R$ and $a$, the Deligne identities in [F1] transport respectively to the integral cup-product identity, the curvature wedge-product identity, and $a(\eta )⌣x=a(\eta \wedge R(x))$. These are conditional deductions; this proof does not establish existence of the Deligne cup-product input.