Let $K$ be a simplicial set and let $\Delta K$ be the category of simplices, i.e the category where the objects are simplicial maps $$ \Delta[n]\to K $$ and the maps $\phi\: : \: (\Delta[n]\to K)\to (\Delta[m]\to K)$ are simplicial maps $\phi\: : \: \Delta[n]\to \Delta[m]$ such that the triangle commute. Now let $L$ be another simplicial set, here my question: is it possible build $\Delta K\times L$ from $\Delta K$, $\Delta L$ in a canonical way?

Here a concrete example: Let $x\: : \:\Delta[n]\to K$, $y\: : \:\Delta[l]\to L$ be two objects of the two categories of simplices. The cartesian products $$ x\times y\: : \:\Delta[n]\times\Delta[l] \to K\times L $$ is not an element of $\Delta (K\times L)$ (for $l,n\neq 0$), well is it possible to find a canonical representant of $x\times y$ inside $\Delta (K\times L)$?

Equivalently: there is a canonical map $\Delta[n]\times\Delta[l]\to \Delta[something]$?