In other words, $\smash {\rho ^{\mathsf{Sets}_{*},\lhd }_{X}}$ acts on elements as
\[ \rho ^{\mathsf{Sets}_{*},\lhd }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (1,x\webright )\webright ] \]
i.e. by
\[ \rho ^{\mathsf{Sets}_{*},\lhd }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\lhd 1 \]
for each $x\in X$.
