Misc. macros:
$\Newextarrow{\xLongrightarrow}{3,2}{0x21D2} \Newextarrow{\xrightrightarrows}{5,5}{0x21C9}$
123, 1\mathop2 3, 1\mathop{2}3, 1{\mathop2}3, 1{\mathop{2}}3
\[123, 1\mathop2 3, 1\mathop{2}3, 1{\mathop2}3, 1{\mathop{2}}3\]
${\bf Span}({\bf Graph})$ $S^{{\mathcal{W}}_\Lambda}\otimes T$
\begin{equation}\begin{CD}
S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\
@VVV @VV{{\rm End}\, P}V\\
(S\otimes T)/I @= (Z\otimes T)/J
\end{CD}\end{equation}
\begin{equation}\begin{CD}
A @>a>> B\\
@VbVV @VVcV\\
C @>d>> D
\end{CD}\end{equation}
A significant
principal $S_n$-bundle ($n \in \bf\text{N}$, $X \in \bf\text{Set}$):
\begin{equation}\begin{CD}
S_{[n]} @>>> \homst {[n]} \Inj X\\
\text{ } @VV\text{image}V\\
\text{} @. \mathcal{P}^{[n]} (X)\\
\end{CD}\end{equation}
An important adjunction (in combinatorics and elsewhere; $X \in \bf\text{Set}$):
\begin{equation}\text{\substack:}\qquad \mathrel{\mathcal{SPart}(X) \substack{\xrightarrow{X/-} \\\xleftarrow[\kernelpart]{}} X\downarrow\Surj} \end{equation}
$$\binom {n} {k} \left\{ \begin{aligned} n\\
k
\end{aligned}
\right\}$$
\buildrel: $\buildrel \text{action associativity} \over =$ and \stackrel: $\stackrel {\text{action associativity}} =$
$$\text{\Newextarrow:}\qquad \Newextarrow{\xequals}{5,5}{0x003D} \xequals{\text{long message}}$$
$$\text{\Newextarrow:}\qquad \Newextarrow{\xequals}{5,5}{0x21C9} \xequals{\text{long message}}$$
$$\text{\xlongequal:}\qquad \xlongequal{\text{long message}}$$
$\Aut X \quad \homst X\Set Y _{subscript1}text_{subscript2}$
$$\text{\atop:}\qquad X {\longrightarrow \atop \longrightarrow} Y$$
$$\hbox{text 1} \qquad \raise10pt{\hbox{text 2}}$$
$\def\graphsoverx{\leftcat{\Set\mathord\downdownarrows}X} \def\setsunderx{X\rightcat{\mathord\downarrow\Set}}$ $\graphsoverx \quad \setsunderx$
\[\begin{array}{}
\Sub (\N) & \xrightarrow{\textstyle \text{least}} & \bftwo^\op \\
& \llap{{?}\notin{?}'} \searrow & \Bigg\Updownarrow \rlap{\mkern-57mu \text{least}{+}{+}PCON} & \searrow \rlap\neg \\
&& [\N,\bftwo] & \xrightarrow[\textstyle PCON]{} & \bftwo \\
\end{array}\]