Working on John's hint, puzzle 155 gives this commuting diagram.
\\[
\begin{matrix}
\text{Lan}_G(H)(\text{Germans}) & \overset{\text{Lan}_G(H)(\text{Friends})}\longrightarrow & \text{Lan}_G(H)(\text{Italians}) \\\\
\alpha\_{\text{Germans}}\downarrow & & \downarrow\alpha\_{\text{Italians}}\\\\
F(\text{Germans}) & \overset{F(\text{Friends})}\longrightarrow & F(\text{Italians})
\end{matrix}
\\]
\\[
\begin{matrix}
\text{Lan}_G(H)(\text{Germans}) & \overset{\text{Lan}_G(H)(\text{Friends})}\longrightarrow & \text{Lan}_G(H)(\text{Italians}) \\\\
\alpha\_{\text{Germans}}\downarrow & & \downarrow\alpha\_{\text{Italians}}\\\\
F(\text{Germans}) & \overset{F(\text{Friends})}\longrightarrow & F(\text{Italians})
\end{matrix}
\\]
Version 2.1.8p2
