Rounded arrow in tikzcd

Question

In tikzcd I would like to draw an arrow with rounded corners which goes from an entry to an entry far right from it, and passes the row below them. It should not just be bent (this works with \ar[bend right=...]), it should rather look like half of a rectangle. Here is what almost works, except that one has to join the two arrows. Also, I would like to put the text "g' \circ g" on it. Probably one has to introduce a phantom node and tell the path to pass through it?

\begin{tikzcd}[column sep=34pt]
B \ar{r}[swap]{g}
\ar[to path= |- (\tikztotarget),rounded corners=12pt]{dr}  & B'
\ar{r}[swap]{g'} & B'' \\
&  \ar[to path= -| (\tikztotarget),rounded corners=12pt]{ru} &
\end{tikzcd}

Edit: In LaRiFari's solution, how can I put text on the arrow? My attempts do not work.

Answer 1

You could do:

% arara: pdflatex

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

\begin{document}
\begin{tikzcd}[column sep=34pt]
    B \ar{r}{g}\ar[to path={ -- ([yshift=-3ex]\tikztostart.south) -| (\tikztotarget) },rounded corners=12pt]{rr}  
    & B' \ar{r}{g'} 
    & B''
\end{tikzcd}
\end{document}