If $\mathcal{L}$ is the $\mathbf{sSet}$-enriched subcategory of $\mathbf{sSet}$ whose objects are finite coproducts of the terminal simplicial set $\Delta^0 = \Delta(-,[0]) = *$, identify the object $\mathcal{L}(*,*)$.
One obvious candidate for $\mathcal{L}(*,*)$ is $\Delta^1$. To that end, we know that $1_{*} = d_o \circ s_0 = d_1 \circ s_0$ by the simplicial identities.
Another (perhaps more likely) canditate for $\mathcal{L}(*,*)$ is $*$ itself, i.e., $\mathcal{L}(*,*) \cong *$. To that end, we know that there exists a unique morphism $1_{\mathcal{L}(*,*)}: \mathcal{L}(*,*) \rightarrow *$, and that there exists as morphism $id: * \rightarrow \mathcal{L}(*,*)$ via the enrichment structure.