Long arrow into a twocell diagram

Question

Through the code:

\documentclass[12pt,a4paper]{memoir}
    \usepackage[all,cmtip,2cell]{xy}
    \UseAllTwocells
    \begin{document}
    \xymatrix@R+20pc@C+3pc{
    \bullet\ruppertwocell^a{\hspace{.40em}ab}
    \rlowertwocell_c{\hspace{.40em}bc}
    \ar[r]_(.40)b & \bullet\\
    }
    \end{document}

I obtain a twocell diagram :

Now I need to have a double arrow named ac from a to c all along the diagram keeping the double arrows ab and bc into the diagram and I don't know how to do it.

Can anybody help with this? Thank you.

Answer 1

If for you is OK to use tikz-cd:

\documentclass[12pt,a4paper]{memoir}

\usepackage{tikz-cd}

\begin{document}
\begin{tikzcd}
    \bullet\ar[r,"b"{name=B, below left},""{name=middle, inner sep=0pt}]\ar[r, bend left=70,"a"{name=A}]\ar[r, bend right=70,"c"{name=C, below}] &[1em] \bullet 
    \arrow[Rightarrow, from=A, to=middle, shorten <=2pt, "ab"]
    \arrow[Rightarrow, from=middle, to=C, shorten =2pt, "bc"]
    \arrow[Rightarrow, from=A, to=C, shorten =2pt, "ac", bend left=100, looseness=3]
\end{tikzcd}
\end{document}

Answer 2

One way is using the Implies arrow tip (in the arrows.meta library) together the double option of TikZ. Change numbers as you like!

\documentclass[tikz,border=5mm]{standalone}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[c/.style={fill,circle,inner sep=2pt,outer sep=2pt},
arr/.style={-{Implies},double,double distance=1mm,shorten >=1mm,shorten <=1mm}] ]
\path 
(0,0) node[c] (X) {}    
(3,0) node[c] (Y) {}
;
\draw[->] (X) to[bend left=60]  node[above,inner sep=2pt] (a) {$a$} (Y);        
\draw[->] (X) to[bend right=60] node[below,inner sep=2pt] (c) {$c$} (Y);
\draw[->] (X)--(Y) coordinate[pos=.5] (M);
\path (M)+(-135:.3)node{$b$};

\draw[arr] (a)--(M) node[pos=.5,right]{$ab$};
\draw[arr] (M)--(c) node[pos=.5,right]{$bc$};
\draw[arr] (a) ..controls +(180:2.5) and +(180:2.5).. (c)  node[pos=.5,left]{$ac$};

\end{tikzpicture}
\end{document}