How can I typeset the included diagram in xymatrix?

Question

\documentclass[10pt]{article}
\usepackage{fancyhdr,graphicx,lastpage}
\usepackage[all]{xy}
\usepackage{listings}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{makeidx}
\makeindex  

\usepackage[a4paper,hmargin=2.5cm,vmargin=2.5cm]{geometry}
\renewcommand{\rmdefault}{ptm}
\renewcommand{\bfdefault}{b}
\usepackage[scaled=0.92]{helvet}
\usepackage{courier}
\normalfont % in case the EC fonts aren't available
\usepackage[T1]{fontenc}
\usepackage{etoolbox}
\makeatletter

\begin{document}
Given categories $\mathbb{A}$ and $\mathbb{B}$ consider the natural transformations:
$\xymatrix{\mathbb{A}\ar@<2ex>[r]^F^{\alpha} \ar[r]_G \ar@<-2ex>[r]_H^{\beta}&\mathbb{B}}$ 
\end{document}$

Please help me typeset this arrow diagram in xymatrix. This is vertical composition of two natural transformations.

Answer 1

I know you requested a solution with xy, but you may want to consider a tikz-cd solution.

Here is the code:

\documentclass{article}
\usepackage{amsfonts}
\usepackage{tikz-cd}
\begin{document}
[\begin{tikzcd}[row sep=.3cm, column sep=3cm]
\phantom{\mathbb{A}}\arrow[r, "F" description, ""{name=top, below}] & \phantom{\mathbb{B}}\
\mathbb{A}\arrow[r,"G" description, ""{name=midtop, above}, ""{name=midbot, below}] & \mathbb{B}\
\phantom{\mathbb{A}}\arrow[r, "H" description, ""{name=bot}] & \phantom{\mathbb{B}}
\arrow[from=top,to=midtop,"\alpha"]
\arrow[from=midbot,to=bot,"\beta"]
\end{tikzcd}]
\end{document}

The idea is to name the space above and below the center labels so that the vertical arrows can be placed correctly. The \phantom symbols are used to keep the horizontal arrows the same length.

You can adjust the spacing to your liking by changing row sep and column sep values.

Answer 2

Updated 1: Using an hacky.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage[all,cmtip]{xy}
\begin{document}
\xymatrix@R=.8pc@C=8pc{
\mathbb{A}  \ar[r]|G  \ar@<-4ex>[r]H \ar@<4ex>[r]^F{\mkern9.8mu\big\downarrow \alpha} \ar@<10.5ex>[d]^{\beta} &\mathbb{B}\
      &
}
\end{document}

---

I'm not very able with the explanations but it not necessary to use double dollars $ with \xymatrix as from my example, that is similar to the original image. My code it is very simple because I'm not an expert (for my low level to understand the documentations) and you can see the minimal differences.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage[all]{xy}
\begin{document}
\xymatrix@1@C=6pc{
\ar[r]^F \ar@<7.9ex>[d]^{\alpha}&\
\mathbb{A} \ar[r]|G \ar@<7.9ex>[d]^{\beta}& \mathbb{B}\
\ar[r]_H & 
}
\end{document}

Here there is a different approaches for the rows R and the columns C.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage[all]{xy}
\begin{document}
xymatrix@R=1pc@C=5pc{
\ar[r]^F \ar@<7.2ex>[d]^{\alpha}&\
\mathbb{A} \ar[r]|G \ar@<7.2ex>[d]^{\beta}& \mathbb{B} \
\ar[r]_H & 
}
\end{document}