Working on the two explanations given above, the natural transformation \$$\beta\_\text{Italians}\$$ maps *every* Italian in \$$F\$$ to *the* Italian in \$$\mathrm{Ran}\_G (H)\$$, of which there can be only one unique possible function.

We can therefore reason that \$$\beta\_\text{Germans}\$$ and \$$\alpha\_\text{Germans}\$$ are effectively the same map, where the sizes of each are equated by
\\begin{align} |\alpha\_\text{Germans}| \\\\ = |F\circ G(\text{Germans})|^{|H(\text{Germans})|} \\\\ = 4^5 \\\\ = |F(\text{Germans})|^{|Ran\_G(H)(\text{Germans})|} \\\\ = |\beta\_\text{Germans}|. \end{align} \