In other words, $\smash {\rho ^{\mathsf{Sets}_{*},\rhd }_{X}}$ acts on elements as
\begin{align*} \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\rhd 0\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0},\\ \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\rhd 1\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\end{align*}
for each $x\rhd 1\in X\rhd S^{0}$.
