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}
\\]