Script for dualizing diagrams in tikzcd by mirroring

Question

My question is very similar to this one, but in some cases I would prefer my diagrams to be mirrored instead of rotated. Is it possible to write a script which achieves this in LaTeX? As an example I have this diagram for the product:

\begin{tikzcd}
C \arrow[rr] \arrow[drr, "\phi_x"] \arrow[ddrr]
\arrow[dd, "\exists! \Phi", dashed] && \vdots \\
&& F(x) \\
\prod F \arrow[uurr] \arrow[urr, "\pi_x", swap] \arrow[rr] && \vdots
\end{tikzcd}

and its dual, the coproduct:

\begin{tikzcd}
\vdots \arrow[rr] \arrow[ddrr] && C \\
F(x) \arrow[urr, "\phi_x"] \arrow[drr, "\mu_x", swap] \\
\vdots \arrow[uurr]\arrow[rr] && \coprod F \arrow[uu, "\exists ! \Phi", dashed]
\end{tikzcd}

I had an attempt at modifying the script in the accepted answer of the linked question, but failed to produce a solution. Thanks!

Answer 1

For simple diagrams like this you only need to set the column sep to a negative value, swap the labels, and revert the arrows. The negative value here was found by looking at the manual. Significantly more complex, where complex means varying column widths, reverted arrows and so on, diagrams may not be dualized that easily. (One can still write styles that revert all relevant original keys to their "dualized" values but the effort for this may not be justified.)

\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{dual/.style={column sep=#1,every label/.append style={swap},
arrows=leftarrow},
dual/.default=-5.5em}
\begin{document}
Original diagram:
\[\begin{tikzcd}
C \arrow[rr] \arrow[drr, "\phi_x"] \arrow[ddrr]
\arrow[dd, "\exists! \Phi", dashed] && \vdots \\
&& F(x) \\
\prod F \arrow[uurr] \arrow[urr, "\pi_x", swap] \arrow[rr] && \vdots
\end{tikzcd}\]
Dual constructed by hand"
\[\begin{tikzcd}
\vdots \arrow[rr] \arrow[ddrr] && C \\
F(x) \arrow[urr, "\phi_x"] \arrow[drr, "\mu_x", swap] \\
\vdots \arrow[uurr]\arrow[rr] && \coprod F \arrow[uu, "\exists ! \Phi", dashed]
\end{tikzcd}\]
Dual constructed by adding the pgf key \texttt{dual} to the original diagram.
\[
\begin{tikzcd}[dual]
C \arrow[rr] \arrow[drr, "\phi_x"] \arrow[ddrr]
\arrow[dd, "\exists! \Phi", dashed] && \vdots \\
&& F(x) \\
\prod F \arrow[uurr] \arrow[urr, "\pi_x", swap] \arrow[rr] && \vdots
\end{tikzcd}\]
\end{document}

How could make this things easier? You could just store the original diagram in a macro (using an appropriate ampersand replacement) and then dualize it reusing the macro.

\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{dual/.style={column sep=#1,every label/.append style={swap},
arrows=leftarrow},dual/.default=-5.5em,ampersand replacement=\&}
\begin{document}
\def\OriginalDiagram{C \arrow[rr] \arrow[drr, "\phi_x"] \arrow[ddrr]
\arrow[dd, "\exists! \Phi", dashed] \&\& \vdots \\
\&\& F(x) \\
\prod F \arrow[uurr] \arrow[urr, "\pi_x", swap] \arrow[rr] \&\& \vdots
}
Original diagram:
\[\begin{tikzcd}
\OriginalDiagram
\end{tikzcd}\]
Dual constructed by adding the pgf key \texttt{dual} to the original diagram.
\[
\begin{tikzcd}[dual]
\OriginalDiagram
\end{tikzcd}\]
\end{document}

Needless to say that this duality does not turn a \pi into a \mu and a \prod into a \coprod.

Answer 2

I figured I could add something to the conversation by noting that you can similarly to the accepted answer set row sep to a negative value to mirror the diagram horizontally. Though the original question described vertical mirroring.

\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{vdual/.style={column sep=#1,every label/.append style={swap},
arrows=leftarrow},
vdual/.default=-5.5em, ampersand replacement=\&}
\tikzcdset{hdual/.style={row sep=#1,every label/.append style={swap},
arrows=leftarrow},
hdual/.default=-5.5em, ampersand replacement=\&}

\begin{document}
\def\OriginalDiagram{C \arrow[rr] \arrow[drr, "\phi_x"] \arrow[ddrr]
\arrow[dd, "\exists! \Phi", dashed] \&\& \vdots \\
\&\& F(x) \\
\prod F \arrow[uurr] \arrow[urr, "\pi_x", swap] \arrow[rr] \&\& \vdots
}
Original diagram:
\[\begin{tikzcd}
\OriginalDiagram
\end{tikzcd}\]

Dual constructed by adding the pgf key \texttt{vdual} to the original diagram.
\[
\begin{tikzcd}[vdual]
\OriginalDiagram
\end{tikzcd}\]

Dual constructed by adding the pgf key \texttt{hdual} to the original diagram.
\[
\begin{tikzcd}[hdual]
\OriginalDiagram
\end{tikzcd}\]
\end{document}

Apologies if I'm not following etiquette.