A Cartesian Promonoidal Kernel on $\mathrm{\Delta }$ and a Hadamard Contraction of ${\mathrm{\Delta }}^n$

We exhibit a symmetric promonoidal kernel on the simplex category $\mathrm{\Delta }$ with Cartesian unit, yielding on representable functors a Hadamard natural transformation ${\mathrm{\Delta }}^p\times {\mathrm{\Delta }}^q\to {\mathrm{\Delta }}^{pq}$ based on pointwise multiplication of nondecreasing maps. Specializing to $q=1$ yields a simplicial homotopy contracting ${\mathrm{\Delta }}^n$ to its $0$-vertex. The contraction is classical; the promonoidal presentation and induced Hadamard map appear not to be recorded.