\[
\tikzcdset{xharp/.style={harpoon,shift left=0.2ex}}
\begin{tikzcd}[column sep=3em]
\ol{\Z_b[\beta]_\f}
\ar[r, "D"]
\ar[d, xharp, "\bm h"]
&
\ol{\Z_b[\beta]_\f}
\ar[d, xharp, "\bm h"]
\\
\AA^\omega
\ar[u, xharp]
\ar[r, "\text{(shift)}"]
\ar[d, xharp, "(*)"]
&
\AA^\omega
\ar[u, xharp]
\ar[d, xharp, "(*)"]
\\
\inverselim \Z[\beta]/\beta^n\Z[\beta]
\ar[u, xharp]
\ar[r, "\text{(shift)}"]
&
\ar[u, xharp]
\inverselim \Z[\beta]/\beta^n\Z[\beta]
\mathrlap,\end{tikzcd}