Beware that, without some qualification, organizing examples of categories is a task like tabulating sand grains on the beach. That attempt of a “database of categories” – by an optimistic soul from the early days of the nLab – failed to take off due to this problem.

What may work is organizing the typical categories used in a small, well-prescribed topic of mathematics. A good example for this is the diagram of categories of topological vector spaces here, which is actually useful.

Similarly, some of the examples you have in #1 might be organized to “typical categories used in linear algebra” etc., and a decent diagram of these, containing maybe a dozen examples, might work well on the page for linear algebra.

But a general page meaning to organize general categories without any restriction to mathematical topics is hopeless – and needless.

]]>Current tex file (surely with some mistakes): (output: https://imgur.com/a/BxcuqGE )

% !TeX engine = lualatex

\documentclass[tikz]{standalone}

\usetikzlibrary{cd, arrows, graphs, graphdrawing}

\usegdlibrary{layered}

\newcommand{\cat}[1]{\ensuremath{\mathbf{#1}}}

\begin{document}

\begin{tikzpicture}[>={Stealth}, rounded corners]

\graph [layered layout, nodes={font=\bfseries}]{

Vect -> Mod -> Ab -> {Grp, CMon -> Mon -> Cat};

Grp -> {Loop, ISGrp, Mon};

SGrp -> SGrpd;

Grp -> Grpd -> {SGrpd, pSet, Cat -> SGrpd -> Set};

{UMag, Loop} -> pSet -> Set;

{{ISGrp, Mon} -> SGrp, {Loop, ISGrp} -> QGrp, {Loop, Mon} -> UMag} -> Mag -> Set;

Field -> CRing -> Ring -> Grp;

CRing -> Ab;

Poset -> Set;

Hilb -> IPS -> Norm -> TVect -> Top -> Set;

Hilb -> Ban -> CMet -> Met -> "Top\textsubscript{Haus}" -> Top;

Ban -> Norm -> Met;

TVect -> Vect;

Set -> Rel;

};

\end{tikzpicture}

\end{document} ]]>