Command for transverse, and not \pitchfork, as used in Guillemin and Pollack

Here's a way to construct the symbol:

\documentclass{article}
\usepackage{amsmath,amssymb}

\newcommand{\transv}{\mathrel{\text{\tpitchfork}}}
\makeatletter
\newcommand{\tpitchfork}{%
  \vbox{
    \baselineskip\z@skip
    \lineskip-.52ex
    \lineskiplimit\maxdimen
    \m@th
    \ialign{##\crcr\hidewidth\smash{$-$}\hidewidth\crcr$\pitchfork$\crcr}
  }%
}
\makeatother

\begin{document}
$f_{s} \transv Z$

$X_{f_{s} \transv Z}$
\end{document}

If you don't plan to use the symbol in sub/superscripts, you may remove the \text around \tpitchfork.

If you'd like to lower the symbol, add something like \raise-0.1ex just before \vbox in the definition of \tpitchfork and adjust the 0.1 to suit.

\newcommand{\tpitchfork}{%
  \raise-0.1ex\vbox{
    \baselineskip\z@skip
    \lineskip-.52ex
    \lineskiplimit\maxdimen
    \m@th
    \ialign{##\crcr\hidewidth\smash{$-$}\hidewidth\crcr$\pitchfork$\crcr}
  }%
}

I was also looking for this and found it in the UTF-8 character library listed as "Pitchfork With Tee Top"

UTF-8 character: ⫚ (Hex code: U+2ADA)

Interestingly though there is also a character next to it called "Transversal Intersection".

UTF-8 character: ⫛ (Hex code: U+2ADB)

My Math teacher uses the former one (as in your question), so it may be a legitimate use of that symbol. Maybe both are acceptable alternatives? I'm not an expert.