Flipping an Arrow in a Cubical Diagram and Crossing Over

Question

So I'm trying to draw a cubical diagram using tikzcd and have been having a persistent issue with the crossing of arrows. Here is the relevant code for reference:

\documentclass[12pt]{standalone}

\usepackage{amssymb}
\usepackage{tikz-cd}
\begin{document}

\begin{center}
\begin{tikzcd}[row sep=scriptsize,column sep=scriptsize,]
\emptyset\ar[rr,hook]\ar[dr,hook] \ar[dd,hook] & & \      
{3\}\ar[dr,hook]\ar[dd,hook] & \\
& \{2\}\ar[rr,hook,crossing over] & & \{2,3\}\ar[dd,hook] & \\
\{1\}\ar[rr,hook]\ar[dr,hook]& & \{1,3\}\ar[dr,hook] & \\
& \{1,2\}\ar[rr,hook]\ar[crossing over, hookleftarrow]{uu}{} & & \  
{1,2,3\} &
\end{tikzcd}
\end{center}

\end{document}

I'm trying to get the front-left downward arrow to be hooked in the same direction as the rest. In the written code it emanates from {1,2} and goes to {2}, because I want it to cross over the back arrow, and writing it the other way would have it be drawn before the arrow it crosses over (and therefore missing the white space on the crossing). My workaround has been to use the hook arrows from bottom to top, but using hookleftarrow to make it go in the opposite direction. Unfortunately, it has caused the arrow "curling" to go the other way. Any ideas to fix it?

I've read the proposed answers on Delayed drawing of arrows in tikz-cd. However, I'm having trouble getting the main one to work (mainly by means of defining \laterarrow). Is there a "flip" command in tikz-cd that lets me flip an arrow across its long axis? Any help is appreciated.

Answer 1

If you like hacky approaches... here you are:

% arara: pdflatex

\documentclass{standalone}
\usepackage{tikz-cd}

\begin{document}
\begin{tikzcd}[%
    ,row sep=scriptsize
    ,column sep=scriptsize
    ]
    \emptyset\ar[rr,hook]\ar[dr,hook] \ar[dd,hook] & & \{3\}\ar[dr,hook]\ar[dd,hook] & \\
    & \{2\}\ar[rr,hook,crossing over]\arrow[hook]{dd} & & \{2,3\}\ar[dd,hook] & \\
    \{1\}\ar[rr,hook]\ar[dr,hook]& & \{1,3\}\ar[dr,hook] & \\
    & \{1,2\}\ar[rr,hook]\ar[crossing over,dash,shorten <= 3mm,shorten >= 3mm]{uu} & & \{1,2,3\} &
\end{tikzcd}            
\end{document}