The morphism $\smash {\rho ^{\mathsf{Sets}_{*},\rhd }_{X}}$ is almost invertible, with its would-be-inverse
\[ \phi _{X}\colon X\to X\rhd S^{0} \]
given by
\[ \phi _{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\rhd 1 \]
for each $x\in X$. Indeed, we have
\begin{align*} \webleft [\rho ^{\mathsf{Sets}_{*},\rhd }_{X}\circ \phi \webright ]\webleft (x\webright ) & = \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (\phi \webleft (x\webright )\webright )\\ & = \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\rhd 1\webright )\\ & = x\\ & = \webleft [\text{id}_{X}\webright ]\webleft (x\webright ) \end{align*}
so that
\[ \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\circ \phi =\text{id}_{X} \]
and
\begin{align*} \webleft [\phi \circ \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webright ]\webleft (x\rhd 1\webright ) & = \phi \webleft (\rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\rhd 1\webright )\webright )\\ & = \phi \webleft (x\webright )\\ & = x\rhd 1\\ & = \webleft [\text{id}_{X\rhd S^{0}}\webright ]\webleft (x\rhd 1\webright ), \end{align*}
but
\begin{align*} \webleft [\phi \circ \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webright ]\webleft (x\rhd 0\webright ) & = \phi \webleft (\rho ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\rhd 0\webright )\webright )\\ & = \phi \webleft (x_{0}\webright )\\ & = 1\rhd x_{0}, \end{align*}
where $x\rhd 0\neq 1\rhd x_{0}$. Thus
\[ \phi \circ \rho ^{\mathsf{Sets}_{*},\rhd }_{X}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{?}}}=}}\text{id}_{X\rhd S^{0}} \]
holds for all elements in $X\rhd S^{0}$ except one.
