[Tex/LaTex] Commutative diagram with rounded arrows

Question

I want to draw a diagram which has some arrows between to points A and B. First arrow goes straight from A to B and above it should be a $\Psi$. Second arrow goes in a round way from A to B and above should be a $\Phi$. Furthermore there's another one from A to B which has a $\Omega$.

Now there should be some $\Downarrow$s from $\Phi$ to $\Psi$ and from $\Psi$ (starting below the arrow) to $\Omega$.

The drawing below shows an idea of what I want.

I usually use xypic for such kind of diagrams. Thatswhy I tried this first. But I came to no real solution. Ist this possible with xypic and, if yes, how? If not, I assume most would suggest TikZ. How can I set this with TikZ?

Answer 1

In Xy-pic:

\documentclass{article}
\usepackage[all,2cell]{xy}
\UseAllTwocells

\begin{document}
\xymatrixcolsep{3cm}
\xymatrix{
  A \ruppertwocell^f{\alpha}
    \rlowertwocell_h{\beta}
    \ar[r]|{g}
  &B\\
}

\end{document}

In TikZ:

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

\begin{document}

\begin{tikzpicture}
  \node (a) {$A$};
  \node[right=4cm of a] (b) {$B$};
  \draw[->]
    (a) edge node[fill=white] (g) {$g$} (b)
    (a) edge[bend left] node (f) [above]{$f$} (b)
    (a) edge[bend right] node (h) [below]{$h$} (b);
  \draw[shorten <=2pt,shorten >=2pt,-implies]
    (f) edge[double] (g)
    (g) edge[double] (h);
\end{tikzpicture}

\end{document}