Natural transformation arrow with TikZ

Question

How do you draw natural transformation arrows as mentioned in this question but using TikZ rather than xy? More generally, is it possible to draw edges between edge labels, from an edge label to a node, and from a node to an edge label or is it necessary to explicitly construct a node in order to draw an edge? Does providing an edge label in the path environment implicitly construct a node that can be used as the source and target of another edge? I would like to do this using the matrix and path environments in TikZ if possible.

Example:

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] {$ H $} (m-2-1);    
\end{tikzpicture}

\end{document}

How can I connect F◦G and H with a curved natural transformation arrow ⟹? Feel free to rearrange the diagram if you'd like.

Answer 1

As percusse mentions in his comment, the tikz-cd package offers you a convenient set of macros to draw commutative diagrams; here's a little example:

\documentclass{article} 
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=huge]
\textbf{CRing}
  \arrow[bend left=50]{r}[name=U,label=above:$\scriptstyle\mathrm{GL}_n$]{}
  \arrow[bend right=50]{r}[name=D,label=below:$\scriptstyle U$]{} &
\textbf{Grp}
  \arrow[shorten <=10pt,shorten >=10pt,Rightarrow,to path={(U) -- node[label=right:$\det$] {} (D)}]{}
\end{tikzcd}

\end{document}

Since originally the question asked for a TikZ solution using a matrix of nodes, here's a "pure" TikZ possible solution:

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzpicture}
\matrix[matrix of nodes,column sep=2cm] (cd)
{
   \textbf{CRing} & \textbf{Grp} \\
};
\draw[->] (cd-1-1) to[bend left=50] node[label=above:$\scriptstyle\mathrm{GL}_n$] (U) {} (cd-1-2);
\draw[->] (cd-1-1) to[bend right=50,name=D] node[label=below:$\scriptstyle U$] (V) {} (cd-1-2);
\draw[double,double equal sign distance,-implies,shorten >=10pt,shorten <=10pt] 
  (U) -- node[label=right:$\det$] {} (V);
\end{tikzpicture}

\end{document}

An answer to the edit to the original question, showing two possibilities (a curved double arrow, and a straight one):

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) -- (h.150);    
\end{tikzpicture}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) to[out=-90,in=180] (h.180);    
\end{tikzpicture}

\end{document}

And here's the corresponding code using tikz-cd:

\documentclass{article} 
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[name=h,swap]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=4pt,Rightarrow,to path={(fg.290) -- (h.175)}]{}
\end{tikzcd}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[swap,name=h]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=3pt,Rightarrow,to path={(fg.290) to[out=-90,in=180] (h)}]{}
\end{tikzcd}

\end{document}

To add a label to the double arrow (as requested in a comment), you can use an additional node; here's an example using both approaches (the first one using tikz-cd and the second one using "pure" TikZ):

\documentclass{article} 
\usepackage{tikz-cd}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[swap,name=h]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=1pt,Rightarrow,to path={(fg.290) to[out=-90,in=180] node[xshift=-3.5mm] {$\tau$} (h)}]{}
\end{tikzcd}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) to[out=-90,in=180] node[xshift=-3.5mm] {$\tau$} (h.180);    
\end{tikzpicture}

\end{document}

Answer 2

I don't respect some initial conditions because I don't understand what it's interesting to use a matrix for two vertices. I think matrix is very useful but only for complex structures.

Here we need two lines to place the main nodes. I used outer sep=4pt to put the arrows at right places. Then I place two nodes in the middle of each arrows. I named these nodes to be able to draw the last double arrow. The option fo this arrow are complex, so I think it's preferable to use a style.

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{arrows}

\begin{document}

\tikzset{dbl/.style={double,
                     double equal sign distance,
                     -implies,
                     shorten >=10pt,
                     shorten <=10pt}}

\begin{tikzpicture} [every node/.style={outer sep=4pt}]
 \node  (A) at (0,0)  {\textbf{CRing}}; 
 \node  (B) at (4,0)  {\textbf{Grp}};

 \draw[->]  (A.north)  to [bend left = 30]  node[above] (C) {$\scriptstyle\mathrm{GL}_n$} (B.north);
 \draw[->]  (A.south)  to [bend right= 30]  node[below] (D) {$\scriptstyle U$}            (B.south);   
 \draw[dbl]   (C)      to                   node[right]     {$\det$}                      (D);
\end{tikzpicture}     

\end{document}

The aim of this method is to be able to scale the figure like I want, now two examples :

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{arrows}

\begin{document}

\tikzset{dbl/.style={double,
                     double equal sign distance,
                     -implies,
                     shorten >=10pt,
                     shorten <=10pt}}

\begin{tikzpicture} [every node/.style={outer sep=4pt},yscale=1.5]
 \node  (A) at (0,0)  {\textbf{CRing}}; 
 \node  (B) at (4,0)  {\textbf{Grp}};

 \draw[->]  (A.north) to [bend left = 30]  node[above] (C) {$\scriptstyle\mathrm{GL}_n$}  (B.north);
 \draw[->]  (A.south) to [bend right= 30]  node[below] (D) {$\scriptstyle U$}             (B.south);   
 \draw[dbl]   (C)     to                   node[right]     {$\det$}                       (D);
\end{tikzpicture}
\begin{tikzpicture} [every node/.style={outer sep=4pt},xscale=1.5,yscale=1.25]
 \node  (A) at (0,0)  {\textbf{CRing}}; 
 \node  (B) at (4,0)  {\textbf{Grp}};

 \draw[->]  (A.north)  to [bend left = 30]  node[above] (C) {$\scriptstyle\mathrm{GL}_n$} (B.north);
 \draw[->]  (A.south)  to [bend right= 30]  node[below] (D) {$\scriptstyle U$}            (B.south);   
 \draw[dbl]   (C)      to                   node[right]     {$\det$}                      (D);
\end{tikzpicture}   
\end{document}

Answer 3

I just wanted to add that it can be very useful to write a macro for that. For example as follows:

\documentclass{article}
\usepackage{tikz-cd}

\newcommand{\nat}[6][large]{%
  \begin{tikzcd}[ampersand replacement = \&, column sep=#1]
    #2\ar[bend left=40,""{name=U}]{r}{#4}\ar[bend right=40,',""{name=D}]{r}{#5}\& #3
          \ar[shorten <=10pt,shorten >=10pt,Rightarrow,from=U,to=D]{d}{~#6}
    \end{tikzcd}
}
\newcommand{\te}[1]{\text{#1}}
\newcommand{\tf}[1]{\textbf{#1}}

The command \nat takes 6 arguments but includes one default argument for the column separation, so you just need to specify 5. Then your document looks as follows:

\begin{document}

    \nat{\tf{CRing}}{\tf{Mon}}{M_n}{U}{\te{det}}
    \nat[huge]{\tf{CRing}}{\tf{Mon}}{M_n}{U}{\te{det}}

\end{document}`