I drew the following diagram. I want to write "j" as a description in the midway (between K and P) but it doesn't apear. My code is:
\documentclass{article}
\usepackage{amsmath,tikz-cd}
\begin{document}
\begin{equation*}
\begin{tikzcd}[row sep=huge,column sep=huge]
0 \arrow{r} & P \arrow{r}{\begin{bmatrix}
p_{2} \\ p_{1}
\end{bmatrix}
}
& B \bigoplus A \arrow{r}{\begin{bmatrix}
\alpha_{2} ~~ -\alpha_{1}
\end{bmatrix}
} & C \\
& P^{\prime} \arrow[ur,"{\begin{bmatrix}
{p^{\prime}}_{2} \\ {p^{\prime}}_{1}
\end{bmatrix}}" description] \arrow[r,dotted,"\exists!~u"'] & K
\arrow[u,tail,"i ~ = ~ \begin{bmatrix}
i_{B} \\ i_{A}
\end{bmatrix}"']
\arrow[ul,"j" description, to path={--
([yshift=-2ex,xshift=-3ex]\tikztostart.south) -|
([yshift=-2ex,xshift=-3ex]\tikztotarget.south) -- (\tikztotarget)}]
\end{tikzcd}
\end{equation*}
\end{document}