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