homotopy theory - An explicit description of Lawvere's segment in the category of simplicial sets

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 $ain A$, $L(a)={xhookrightarrow h_a: xin Ob([A^{op},Set])}$.

That is, $L$ assigns to each object $ain Ob(A)$ 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 $ain A$, 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 (as opposed to proper class) of arrows $Ssubset Arr([A^{op},Set])$, 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, that is to say, geometrically?

Edit: A cellular model M on a presheaf topos is a set of monomorphisms $M$ such that $llp(rlp(M))$ is the class of all monomorphisms. Note that $llp$ and $rlp$ give the class of arrows with the left lifting property (resp. right lifting property) to the class of arrows given in the argument.