It is intuitively clear that a prism $\Delta ^n\times \Delta ^1$ can be triangulated. If I am interpreting this correctly, it should be possible to give a formula for the prism as a colimit (in the diagram of simplicial sets) of some diagram consisting of copies of $\Delta ^{n+1}$.
Is this possible? What are the formulas?
Let $X = \Delta^n \times \Delta^1$. Let $U$ be the disjoint union of the $n+1$ simplices of $X$. The inclusions give a canonical map $U \to X$. This is an epimorphism (why?).
In sSet (or any presheaf category, or even topos), epimorphisms are given by equivalence relations: there is a coequalizer diagram
$$ U \times_X U \rightrightarrows U \to X $$
This decomposes into the individual pullbacks. Since all of the inclusions are monic, we can simplify this to a coequalizer diagram
$$ \coprod_{i<j} (s_i \cap s_j) \rightrightarrows \coprod_{i} s_i \to X $$
where $s_i : \Delta^{n+1} \to X$ are the individual simplices.
If you really want all of the objects in the diagram to be copies of $\Delta^{n+1}$, you can do this in two steps:
To help with all of this, recall that the nerve construction makes Poset a full subcategory of sSet, and furthermore the inclusion preserves coproducts all limits. Furthermore, $\Delta^k$ is the nerve of the poset $[k]$
All of the above can actually be carried out in Poset rather than sSet, and this fact is almost entirely due to the properties of the nerve.
The only place where we need to used the fact we are working in sSet is to argue that $U \to X$ is an epimorphism, and thus $X$ is computed by this coequalizer.
More generally, there is a systematic way to write presheaves over any small category as a colimit of representables whose source diagram is its category of elements. In the case of simplicial sets, the full subcategory spanned by the elements that are nondegenerate simplices is final, so you can restrict the source diagram to that subcategory.