\newcommand{\subsubsubsection}[1]{\textbf{#1}}
\newcommand{\WIP}{\textbf{WIP}}

\newcommand{\coloneqq}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
                     \hbox{\scriptsize.}\hbox{\scriptsize.}}}%
  =}

\newcommand{\defeq}{\mathrel{\triangleq}}
%% Alternatively:
%% \newcommand{\defeq}{≔}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bX}{\mathbb{X}}
% \newcommand{\to}{\rightarrow}
\newcommand{\mto}{\mapsto}
\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace}
\let\type\UU
\newcommand{\MCU}{\UU}
\newcommand{\nomen}[1]{\emph{#1}}
\newcommand{\todo}[1]{\textit{#1}}
\newcommand{\comp}{\circ}
\newcommand{\x}{\times}
\newcommand\inv[1]{#1\raisebox{1.15ex}{$\scriptscriptstyle-\!1$}}
\newcommand{\tp}{\mathrel{:}}
\newcommand{\Type}{\mathcal{U}}

\usepackage{graphicx}
\makeatletter
\newcommand{\shorteq}{%
  \settowidth{\@tempdima}{-}% Width of hyphen
  \resizebox{\@tempdima}{\height}{=}%
}
\makeatother
\newcommand{\var}[1]{\ensuremath{\mathit{#1}}}
\newcommand{\Hom}{\var{Hom}}
\newcommand{\fmap}{\var{fmap}}
\newcommand{\bind}{\var{bind}}
\newcommand{\join}{\var{join}}
\newcommand{\omap}{\var{omap}}
\newcommand{\pure}{\var{pure}}
\newcommand{\idFun}{\var{id}}
\newcommand{\Sets}{\var{Sets}}
\newcommand{\Set}{\var{Set}}
\newcommand{\hSet}{\var{hSet}}
\newcommand{\id}{\var{id}}
\newcommand{\isEquiv}{\var{isEquiv}}
\newcommand{\idToIso}{\var{idToIso}}
\newcommand{\isSet}{\var{isSet}}
\newcommand{\isContr}{\var{isContr}}
\newcommand{\isGroupoid}{\var{isGroupoid}}
\newcommand{\pathJ}{\var{pathJ}}
\newcommand\Object{\var{Object}}
\newcommand\Functor{\var{Functor}}
\newcommand\isProp{\var{isProp}}
\newcommand\propPi{\var{propPi}}
\newcommand\propSig{\var{propSig}}
\newcommand\PreCategory{\var{PreCategory}}
\newcommand\IsPreCategory{\var{IsPreCategory}}
\newcommand\isIdentity{\var{isIdentity}}
\newcommand\propIsIdentity{\var{propIsIdentity}}
\newcommand\IsCategory{\var{IsCategory}}
\newcommand\Gl{\var{\lambda}}
\newcommand\lemPropF{\var{lemPropF}}
\newcommand\isPreCategory{\var{isPreCategory}}
\newcommand\congruence{\var{cong}}
\newcommand\identity{\var{identity}}
\newcommand\isequiv{\var{isequiv}}
\newcommand\qinv{\var{qinv}}
\newcommand\fiber{\var{fiber}}
\newcommand\shuffle{\var{shuffle}}
\newcommand\Univalent{\var{Univalent}}
\newcommand\refl{\var{refl}}
\newcommand\isoToId{\var{isoToId}}
\newcommand\rrr{\ggg}
\newcommand\fish{\mathrel{\wideoverbar{\rrr}}}
\newcommand\fst{\var{fst}}
\newcommand\snd{\var{snd}}
\newcommand\Path{\var{Path}}
\newcommand\Category{\var{Category}}
\newcommand\TODO[1]{TODO: \emph{#1}}
\newcommand*{\QED}{\hfill\ensuremath{\square}}%
\newcommand\uexists{\exists!}
\newcommand\Arrow{\var{Arrow}}
\newcommand\NTsym{\var{NT}}
\newcommand\NT[2]{\NTsym\ #1\ #2}
\newcommand\Endo[1]{\var{Endo}\ #1}
\newcommand\EndoR{\mathcal{R}}