On the image of the almost strict Morse n-category under almost strict n-functors

In an earlier work, we constructed the almost strict Morse $n$-category $\mathcal{X}$ which extends Cohen $\mathrm{\&}$ Jones $\mathrm{\&}$ Segal’s flow category. In this article, we define two other almost strict $n$-categories $\mathcal{V}$ and $\mathcal{W}$ where $\mathcal{V}$ is based on homomorphisms between real vector spaces and $\mathcal{W}$ consists of tuples of positive integers. The Morse index and the dimension of the Morse moduli spaces give rise to almost strict $n$-category functors $\mathcal{F}:\mathcal{X}\to \mathcal{V}$ and $\mathcal G : \mathcal X \to \mathcal W$.