Support for multimap arrow head
tikzcd allows for the "-multimap" option, which sets the arrow head to a circle/lollipop. There is room in the arrow options toolbar of quiver for this option, I think.
As far as I'm aware, tikz-cd does not support multimap by default. If I'm mistaken, could you provide a minimal working example demonstrating it? It could still be possible to support multimaps in quiver, but it would require changes to quiver.sty.
You are right, I hadn't realised this was a macro created (or at least imported) by a coauthor. Here is a MWE:
\documentclass{article}
\usepackage{amsmath}
\usepackage{txfonts}
\usepackage{tikz-cd}
\newcommand{\oc}{!}
\tikzset{%
multimap/.tip={Glyph[glyph math command=multimap]},
}
\begin{document}
\begin{equation*}
\begin{tikzcd}
\oc E \ar[-multimap, r, "\oc \Delta_E"] \ar[-multimap,d,"e"]
& \oc(E \times E) \ar[dr, -multimap, "\cong"', "\chi_{E,E}^{-1}"] \\
X & & \oc E \otimes \oc E
\end{tikzcd}
\end{equation*}
\end{document}
Packages other than txfonts can also provide multimap. Another option once you have the tutorial running, if you don't want to modify quiver or want to save on future modifications, would be to include a "custom" option in quiver with instructions on how to write the macro that this will be replaced with.