Any non-decreasing sequence of naturals can be identified with a functor \(\mathbb{N}\rightarrow \mathbb{N}\), considering the preorder of naturals as a category.
A natural transformation between two of these functors would require a component \(\alpha_n\) for each natural, which means a morphism from \(F_n \rightarrow G_n\). This exists iff \(F(n)\leq G(n)\).
Thus we can put a preorder structure over the monotone map of \(\mathbb{N} \rightarrow \mathbb{N}\) (this is a thin functor category \(\mathbb{N}^\mathbb{N}\)).