Represent schematically using the tikz package a "commutative" diagram where two of the "arrows" are circular loops

I propose to use a scaled version of \circlearrowleft on a phantom vertical arrow:

\documentclass{article}
\usepackage{amssymb, mathrsfs}
\usepackage{adjustbox}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[row sep=large, column sep=large]
{\mathscr{D}_{n,k}[t]} \arrow[rr, "t\mapsto s-m"'] \arrow[rr, phantom, shift left=1ex, swap, "\sim"]
\arrow[d, phantom, "\adjustbox{scale={2}{-2}}{$ \circlearrowleft $}" description]&& {\mathscr{D}_{n,k}[s]} \arrow[d, phantom, "\adjustbox{scale={2}{-2}}{$ \circlearrowleft $}" description] \\
M_{f^t} \arrow[rr, "{g(t,x)\cdot f^t \mapsto g(s-m,x)\cdot f^{s-m}}"'] \arrow[rr, phantom, shift left=1ex, swap, "\sim"]&& M_{f^s}
\end{tikzcd}

 \end{document} 

Updating 1

....with some modifications.....to have a symmetrical loop.

\documentclass[12pt]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{mathtools}
\usepackage{amssymb,mathrsfs}
\usepackage{graphicx}
\usepackage{tikz-cd} 
\newcommand{\cir}{\rotatebox[origin=c]{180}{$\circlearrowright$}}
\begin{document}

\[\begin{tikzcd}[row sep=0em, column sep=4.3em]
\underset{\scalebox{2}{\ensuremath{\cir}}}{\mathscr{D}_{n,k}[t]} \arrow[rr, above, "\sim", "t\mapsto s-m"'] &  & \underset{\scalebox{2}{\ensuremath{\cir}}}{\mathscr{D}_{n,k}[s]}\\[-.5em]
M_{f^t}\arrow[rr, above, "\sim", "{g(t,x)\cdot f^t \mapsto g(s-m,x)\cdot f^{s-m}}"'] &  & M_{f^s}
\end{tikzcd}\]

\end{document}

..... the final code ....

\documentclass[12pt]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{mathtools}
\usepackage{amssymb,mathrsfs}
\usepackage{graphicx}

\usepackage{tikz-cd} 
\newcommand{\cir}{\rotatebox[origin=c]{180}{$\circlearrowright$}}
\begin{document}


\[\begin{tikzcd}[row sep=-.1em, column sep=4.3em]
\underset{\scalebox{3}{\ensuremath{\cir}}}{\mathscr{D}_{n,k}[t]} \arrow[rr, above, "\sim", "t\mapsto s-m"'] &  & \underset{\scalebox{3}{\ensuremath{\cir}}}{\mathscr{D}_{n,k}[s]}\\
M_{f^t}\arrow[rr, above, "\sim", "{g(t,x)\cdot f^t \mapsto g(s-m,x)\cdot f^{s-m}}"'] &  & M_{f^s}
\end{tikzcd}\]


\end{document}

Update 1: new loop my loop

as requested by Anush (I hope this will be convenient). The style of the loops is defined with:

\tikzset{my loop/.style={to path={.. controls +(130:2) and +(50:2) .. (\tikztotarget)}}}

\documentclass[border=5mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{tikz-cd}
\usepackage{mathrsfs}

\tikzset{my loop/.style={to path={.. controls +(130:2) and +(50:2) .. (\tikztotarget)}}}

\begin{document}

\begin{tikzcd}[row sep=large, column sep=large,]
{\mathscr{D}_{n,k}[t]} \arrow[rr,"\sim", "t\mapsto s-m"'] && {\mathscr{D}_{n,k}[s]}  \\
|[alias=Z]|M_{f^t} \arrow[rr, "\sim","{g(t,x)\cdot f^t \mapsto g(s-m,x)\cdot f^{s-m}}"'] && M_{f^s}      
\arrow[my loop, from=2-1, to=2-1
]
\arrow[my loop, from=2-3, to=2-3]
\end{tikzcd}

\end{document}

Old answer

A solution that uses TikZ's native loops. The style of the loops is defined with: :

\tikzset{every loop/.style ={out=120,in=60,looseness=6,min distance=7mm}}

\documentclass[border=5mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{tikz-cd}
\usepackage{mathrsfs}

\tikzset{every loop/.style ={out=120,in=60,looseness=6,min distance=7mm}}

\begin{document}
%\[
\begin{tikzcd}[row sep=large, column sep=large,]
{\mathscr{D}_{n,k}[t]} \arrow[rr,"\sim", "t\mapsto s-m"'] && {\mathscr{D}_{n,k}[s]}  \\
M_{f^t} \arrow[rr, "\sim","{g(t,x)\cdot f^t \mapsto g(s-m,x)\cdot f^{s-m}}"'] && M_{f^s}      
\arrow[loop, from=2-1, to=2-1]
\arrow[loop, from=2-3, to=2-3]
\end{tikzcd}
%\]
\end{document}