The Leibniz PROP is a crossed presimplicial algebra

We prove that the Leibniz PROP is isomorphic (as $\mathbb{k}$-linear categories) to the symmetric crossed presimplicial algebra $\mathbb{k}[({\mathrm{\Delta }}^{+}{)}^{op}\mathbb{S}]$ where ${\mathrm{\Delta }}^{+}$ is the skeletal category of finite well-ordered sets with surjections, but the distributive law between $({\mathrm{\Delta }}^{+}{)}^{op}$ and the symmetric groups $\mathbb{S}={⨆}_{n\ge 1}{S}_n$ is not the standard one.