Typesetting a diagram with "tree arrows"

Question

I'm trying to tex a diagram of the following form:

Even more preferred, I'd like to both have labels for the trees (like $\mathcal T$ and $\pi\mathcal T$ in the above picture), and also a label on the solid arrow sticking out from the tree.

Can this be done in xymatrix, or do I have to get the big tikz guns out?

Thanks!

(this specific diagram is from John Steel's article "An outline of inner model theory", by the way)

Answer 1

This is an examply with tikz-cd. The code for the half dotted arrow is slightly modified from this answer by Ignasi, also including some additions.

Output

Code

\documentclass[margin=10pt]{standalone}
\usepackage{tikz-cd}

\usetikzlibrary{decorations.pathreplacing,calc,arrows.meta}

\tikzset{
    triple/.style={
    decoration={show path construction, 
      lineto code={
          \draw[dotted,-] (\tikzinputsegmentfirst) --($(\tikzinputsegmentfirst)!.7!(\tikzinputsegmentlast)$) coordinate (a);,
          \draw[-{Latex}] ($(\tikzinputsegmentfirst)!.7!(\tikzinputsegmentlast)$)--(\tikzinputsegmentlast);
          \draw[-] (\tikzinputsegmentfirst) --++ (10:1.6cm) node[midway, above, font=\footnotesize] {#1};
          \draw[-] (\tikzinputsegmentfirst) --++ (-10:1.6cm);,
      }
    },
    decorate
    },
}

\begin{document}
\begin{tikzcd}
N \arrow[rr, triple={$\pi T$}]& & N_{\alpha}\\
M \arrow[u, swap, "\pi"] 
\arrow[rr, triple={$T$}]
& & M_{\alpha} \arrow[u, swap, "\pi_{\alpha}"]\\
\end{tikzcd}
\end{document}