I have quite a few functions $F_i:\mathbb{R}^n\to\mathbb{R}^m$, where $n$ and $m$ aren't the same for each $F_i$. e.g. $F_0:\mathbb{R}^2\to\mathbb{R}^2$, $F_2:\mathbb{R}^3\to\mathbb{R}^4$, etc.
I'm trying to write a composition of some number of $F_i$, say:
$F_5 \circ F_4 \circ F_3 \circ F_2 \circ F_1 \circ F_0 \left(\vec{x}\right)$
I'm wondering if there's a standard/agreed upon way to express the input and output dimensions at each stage of the function composition, something like $F_i^{n\to m}$? It feels clunky to write out the composition and then enumerate "where $F_0:\mathbb{R}^2\to\mathbb{R}^2$, $F_1:\cdots$, etc."
If it helps at all, I'm trying to express an ANN, where each $F_i$ represents the transformation from one layer to another. I thought function composition would be a useful way to reinforce the fact that an ANN is just a nonlinear function (ANNs are not really used in my field so I want to make sure the reader "believes" in my methodology). I thought showing the input and output space could also reinforce the concept that each layer is composed of some number of nodes, but am struggling to find a convenient notation online that does all of this.