In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in Ob([A^{op},Set])\}$.

That is, $L$ assigns to each object $a\in Ob(A)$ the set of monomorphisms with target $h_a(\cdot):=Hom(\cdot,a)$
in $[A^{op},Set]$. Since all objects in $[A^{op},Set]$ can be given as colimits of the representable functors $h_a$ for $a\in A$, we can actually give a characterization of the functor represented by $L$ as the "sub-object assigning functor".

Lawvere's segment is useful, because it naturally has the structure of a cylinder functor (given by taking the cartesian product with a presheaf $X$). In fact, it is a theorem of Cisinski (very closely related to the specialization Jeff Smith's theorem to presheaf toposes) that every accessible localizer on a presheaf category admits the structure of a closed model category generated by the homotopical data (donnée homotopique is the term used by Cisinski, so this is my rough translation) of some set $S$, some cellular model $\mathcal{M}$, and the canonical cylinder given by Lawvere's segment.

Then my question: How can we describe Lawvere's segment in $Set_\Delta$ explicitly?