tikz diagram question

Question

I want to create the diagram in the pic below

and I have the code for the upper part of it, but I have hard time to finish the rest, any help appreciated.

my code:-

\documentclass[11pt, oneside]{article}      % use "amsart" instead of "article" for AMSLaTeX format
\usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                          % ... or a4paper or a5paper or ... 
\usepackage{graphicx}
\usepackage{tikz-cd}
\usetikzlibrary{decorations.markings}
\tikzset{middleArrowHead/.style={decoration={markings, mark= at position #1 with \arrow{>}}, postaction=decorate}}
\usepackage{braket} % defines \ket and \bra
\begin{document}
\begin{tikzcd}[every arrow/.append style={bend left, dash, middleArrowHead=1}] % change the last value until the arrow position pleases you.
\mathcal{B}^{'}
\arrow[start anchor={[yshift=1.5pt]east},end anchor={[yshift=2.5pt]west}]{r}{U^{-1}AU^{-1}} 
& \mathcal{C}^{'}
\arrow[start anchor={[yshift=-1.5pt]west},end anchor={[yshift=-2.5pt]east}]{l}{U^{-1}A^{-1}U}
\end{tikzcd}

\end{document}

Answer 1

I'm feeling kind tonight, so here's a complete solution. Usually we try to have you do most of the work and get as close as you can to the solution, because that is the best way to learn a new skill, trying it our yourself. Here's the code:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\tikzset{>={Straight Barb[length=4pt]}}

\begin{document}

\begin{tikzpicture}[scale=3]
% Nodes
\node (b0) at (0,0) {$\mathcal B$};
\node (c0) at (1,0) {$\mathcal C$};
\node (b1) at (0,1) {$\mathcal B'$};
\node (c1) at (1,1) {$\mathcal C'$};
% Arrows
\draw[->] (b1) to [out=40,in=140] node[above] {$U^{-1}AU$} (c1);
\draw[->] (c1) to [out=-140,in=-40] node[below] {$UA^{-1}U$} (b1);
\draw[->,dashed,red] (b0) to node[left,black] {$U^{-1}$?} (b1);
\draw[->,dashed,red] (c0) to node[right,black] {$U$?} (c1);
\end{tikzpicture}

\end{document}

And here's the result:

I'm not a big fan of the tikz-cd environment, I like tikzpicture better, since it's more versatile for general diagrams. Maybe someone else has a better answer in tikz-cd.

Answer 2

Here is a code with tikz-cd. I added a solution with pstricks:

\documentclass{article}
\usepackage{pst-node}
\usepackage{auto-pst-pdf} \usepackage{tikz-cd}

\begin{document}
%
\[ \psset{arrows=->, arrowinset=0.12, linewidth=0.5pt, arcangle = 40, , nodesep=3pt, rowsep=1.2cm, colsep = 1.5cm, shortput =tablr}
  \everypsbox{\scriptstyle}
  \begin{psmatrix}
    \mathcal B' & \mathcal C'\\%
    \mathcal{B} & \mathcal{C} %
%%%%%% arrows%
 {\psset{nodesepA=2pt, nodesepB=1pt, labelsep=-2pt}
      \ncarc{1,1}{1,2}^{U^{-1}AU} \ncarc{1,2}{1,1}_{UA^{-1}U}}
    \psset{linestyle=dashed, dash=3pt 2pt, labelsep=2pt, linecolor=red}
    \ncline{2,2}{1,2}>[l]{u?}
    \ncline{2,1}{1,1}<[r]{u^{-1}?}
  \end{psmatrix}
\]

\[ \begin{tikzcd}[row sep=1.2cm, column sep=1.2cm, inner sep=10ex]
  \mathcal B' \rar[bend left, "U^{-1}AU"] & \mathcal C'\lar[bend left, "UA^{-1}U"] \\%
  \mathcal{B}\uar[dashed, draw=red, "u^{-1}?"] & \mathcal{C}\uar[dashed, draw=red, swap, "u?"] \end{tikzcd}
\]
\end{document}