LaTeX code for commutative diagram

Question

Kindly provide me LaTeX code for this and if possible the circles in each of the boxes saying the diagram is commutative.

For one commutative diagram I have the code

\xymatrixrowsep{10mm}
\xymatrixcolsep{10mm}
\xymatrix{
  \mathcal{F}(U) \ar[r] & \mathcal{G}(U) \\
  \mathcal{F}(V) \ar[u] \ar[r] & \ar[u] \mathcal{G}(V)
}

This is not the code for any part of above diagram, but I am comfortable with this kind of code.

Please see if you can give code which generalizes the code that I have.

That script letters are:

\mathcal{F},\mathcal{G}.

Answer 1

Just make a matrix with four rows, and empty cells where appropriate.

\documentclass{article}
\usepackage[all]{xy}

\begin{document}
\xymatrixrowsep{2mm} % I modified this
\xymatrixcolsep{10mm}
\xymatrix{
\mathcal{F}(U) \ar[dd] \ar[dr] & & \mathcal{F}(V) \ar[ld] \ar[dd]\\
 & \mathcal{F}(W) \ar[dd] & \\
 \mathcal{G}(U) \ar[dr]  &  & \mathcal{G}(V) \ar[ld]\\ 
 & \mathcal{G}(W) &
}
\end{document}    

As egreg mentions in a comment, you can use the cmtip package option to get arrow tips like standard Computer Modern arrows, and \xymatrix@R=2mm@10mm{..} to keep row and column spacing settings local to the diagram:

\documentclass{article}
\usepackage[all,cmtip]{xy} % cmtip tp get standard computer modern arrow tips
\begin{document}
\xymatrix@R=2mm@C=10mm{% keep row and column space local to this diagram
\mathcal{F}(U) \ar[dd] \ar[dr] & & \mathcal{F}(V) \ar[ld] \ar[dd]\\
 & \mathcal{F}(W) \ar[dd] & \\
\mathcal{G}(U) \ar[dr]  &  & \mathcal{G}(V) \ar[ld]\\ 
 & \mathcal{G}(W) &
}
\end{document}  

Answer 2

Here is a solution using the package tikz-cd.

\documentclass[border=1mm,varwidth]{standalone}
\usepackage{tikz-cd}
\usepackage{mathrsfs}
\begin{document}
\paragraph{Step 1:} Position the nodes in a matrix

\begin{tikzcd}
\mathscr{F}(U)   &                & \mathscr{F}(V) \\
                 & \mathscr{F}(W) &                \\
\mathscr{G}(U)   &                & \mathscr{G}(V) \\
                 & \mathscr{G}(W) &                
\end{tikzcd}

\paragraph{Step 2:} Add the arrows

\begin{tikzcd}
\mathscr{F}(U) \arrow{dd}{}\arrow{rd}{}
                 &                & \mathscr{F}(V) \arrow{ld}{}\arrow{dd}\\
                 & \mathscr{F}(W) \arrow{dd}{}
                                  &                \\
\mathscr{G}(U) \arrow{rd}{}
                 &                & \mathscr{G}(V) \arrow{ld}{} \\
                 & \mathscr{G}(W) &                
\end{tikzcd}
\end{document}