A Cartesian Promonoidal Kernel on $Δ$ and a Hadamard Contraction of $Δ^n$

Pointwise multiplication of nondecreasing maps defines simplicial maps \(H_{p,q}:Δ^p\timesΔ^q\toΔ^{pq}\) for nonnegative integers \(p,q\). We express these maps using Day's promonoidal construction for the Cartesian product of presheaves, applied to the simplex category. For \(q=1\), the map is a si