I'm doing this exercise:
Prove that $\mathcal{S}[\beta]$ is a bijection.
Here, $\beta:M\rightarrow N$ is a bijection of finite sets, $\mathcal{S}$ is a species, and $\mathcal{S}[\beta]:\mathcal{S}[M]\rightarrow\mathcal{S}[N]$ is "the induced map" by $\mathcal{S}$.
My problem is that I am having trouble writing what a general $\sigma\in\mathcal{S}[M]$ looks like. I understand how I would prove what is to be shown when $\mathcal{S}$ is the species of finite graphs, the power set, the permutations, or any of the other specific cases of species that I've seen. I understand completely how these work. However, these species all have their particular notation where I can go in and apply $\beta$ to elements of $M$ to show how $\beta$ acts on $\sigma\in\mathcal{S}[M]$, and thus describe the induced map $\mathcal{S}[\beta]$. But how do I do this when $\mathcal{S}$ is kept general? How would I notate $\mathcal{S}[\beta]$ well enough to prove it's bijective?
Your book certainly requires that $\mathcal{S}$ carries composite functions to composites, in the sense that if $M,M',M''$ are finite sets and $M\xrightarrow{f}M'\xrightarrow{g}M''$ are maps between them, then $$\mathcal{S}[g\circ f]=\mathcal{S}[g]\circ\mathcal{S}[f]$$ and furthermore it should carry identity maps to identity maps, in the sense that for any finite set $M$ $$\mathcal{S}[\mathrm{id}_M]=\mathrm{id}_{\mathcal{S}[M]}$$
Now suppose $\beta:M\to N$ is a bijection of finite sets. Apply the two points above to $\beta$ and its inverse map $\alpha$.