\documentclass[a4paper]{memoir}
\usepackage[danish]{babel}
\usepackage{empheq,tikz-cd,amssymb}
\usetikzlibrary{babel}
\begin{document}
\begin{empheq}{equation*}
\begin{tikzcd}[ampersand replacement=\&]
A \arrow[r]
\arrow{d}{f} % works
\& B
%\arrow[d,"f"] % fails
\\
\mathbb{R}
\end{tikzcd}
\end{empheq}
\end{document}