Define ${\cal L}$ as in this question: the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at most finitely many points, and $f\prec g$ is defined to mean $f(n)\leq g(n)$ for all but at most finitely many exceptions $n$.

Let ${\cal P}(\omega)/fin$ be defined as in this post. In another post, it was established that there is some kind of "natural" (not in the categorical sense) lattice embedding from ${\cal P}(\omega)/fin$ into ${\cal L}$.

My question is: Can the relationship between ${\cal P}(\omega)/fin$ and ${\cal L}$ be made more precise? Isn't ${\cal L}$ essentially ${\cal P}(\omega)/fin$ "stacked on top of itself $\omega$ times"? Or is ${\cal L}$ (a quotient of) a product of ${\cal P}(\omega)/fin$?