Homotopy diagram

\documentclass{article}

\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=5mm, row sep=2mm]
\arrow[r]&L^{-i}\arrow[r]&Y\oplus L^{-i}\oplus L^{-i+1}\arrow[r]\arrow[ddddl]& Y\oplus L^{-i+1}\oplus L^{-i+2}\arrow[r]\arrow[ddddl,start anchor={[xshift=-6mm]}]& L^{-i+2}\oplus L^{-i+3}\arrow[r]\arrow[ddddl,start anchor={[xshift=-6mm]}]& L^{-i+3}\oplus L^{-i+4}\arrow[r]\arrow[ddddl,start anchor={[xshift=-6mm]}]&\cdots\\
&&(y,l,l')\arrow[mapsto,ddl,start anchor={[xshift=2mm]},end anchor={[shift={(13mm,2mm)}]}]&(y,l,l')\hspace{16mm}\arrow[mapsto,ddl,start anchor={[xshift=-2mm]},end anchor={[shift={(16mm,2mm)}]}]&(l,l')\hspace{15mm}\arrow[mapsto,ddl,start anchor={[xshift=-2mm]},end anchor={[shift={(15.5mm,2mm)}]}]&(l,l')\hspace{15mm}\arrow[mapsto,ddl,start anchor={[xshift=-2mm]},end anchor={[shift={(14mm,2mm)}]}]\\
\\
&\lefteqn{\hspace{11mm}\hspace{1mm}l}&\lefteqn{\hspace{11mm}(y,0,l)}&\lefteqn{\hspace{10mm}(0,0,l)}&\lefteqn{\hspace{10mm}(0,l)}\\
\arrow[r]&L^{-i}\arrow[r]&Y\oplus L^{-i}\oplus L^{-i+1}\arrow[r]& Y\oplus L^{-i+1}\oplus L^{-i+2}\arrow[r]& L^{-i+2}\oplus L^{-i+3}\arrow[r]&\lefteqn{\cdots}\phantom{ L^{-i+3}\oplus L^{-i+4}}
\end{tikzcd}

\end{document}