Adjusting loop (identity) arrow in tikzcd

Question

With the following code

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

\begin{document}

\begin{tikzcd}
\arrow[loop left]{l}{\mathrm{id}_A} \ar{r}A &
B\times C \arrow[loop right]{r}{\mathrm{id}_{B\times C}} 
\end{tikzcd}

\end{document}

I get:

I'd like to fix the arrows, so that they have the same size independent of the source object, with preference given to the big arrow.

My attempts so far had been only somehow not very effective:

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

\begin{document}

\begin{tikzcd}
\arrow[loop left]{l}{\mathrm{id}_A} \ar{r}A \hspace{.5cm}& 
 {B\times C\arrow[loop right]{r}{\mathrm{id}_{B\times C}} }
\end{tikzcd}

\begin{tikzcd}
\arrow[loop left]{l}{\mathrm{id}_A} \ar{r}A &
\hspace{-.7cm}{B\times \hspace{-1.9cm}}& {C\arrow[loop right]{r}{\mathrm{id}_{B\times C}} }
\end{tikzcd}

\end{document}

Answer 1

I fear, you will have to set the start and end anchors manually. I can't think of a better solution:

% arara: pdflatex

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

\begin{document}    
    \begin{tikzcd}[cells={nodes={}}]
        \arrow[loop left, distance=3em, start anchor={[yshift=-1ex]west}, end anchor={[yshift=1ex]west}]{}{\mathrm{id}_A} \arrow{r} A 
        & B\times C\arrow[loop right, distance=3em, start anchor={[yshift=1ex]east}, end anchor={[yshift=-1ex]east}]{}{\mathrm{id}_{B\times C}} 
    \end{tikzcd}
\end{document}

For even more control on the arrow, you can define [out=-150, in=150, loop, distance=3em,...].