\documentclass[a4peper,12pt]{article}
\usepackage{amsthm, amssymb, amsmath}
\usepackage{tikz}


\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\cone}{cone}
\begin{document}
\begin{center}
\tikzset{node distance=2cm, auto}
\begin{tikzpicture}[line width=.5pt]%[style={very thick}]
\node(a1) {$0$}; 
\node(a2)[right of=a1]{$K'$}; 
\node(a3)[right of=a2]{$A'$}; 
\node(a4)[right of =a3] {$X$}; 
\node(a5)[right of=a4] {$0$};
\node(b1)[below of=a1] {$0$};
\node(b2)[right of=b1]{$K$}; 
\node(b3)[right of=b2]{$A$}; 
\node(b4)[right of=b3]{$X$};
\node(b5)[right of=b4]{$0$};
\node(c1)[below of=b2]{$N$};
\draw[->] (a1)--(a2);
\draw[->] (a2) to node[above] {$f'$} (a3);
\draw[->] (a3) to node[above] {$g'$} (a4);
\draw[->] (a4)--(a5);
\draw[->] (b1)--(b2);
\draw[->] (b2) to node[below] {$f$} (b3);
\draw[->] (b3) to node[below] {$g$} (b4);
\draw[->] (b4) -- (b5);
\draw[double distance=2pt] (b4) -- (a4);
\draw[->] (a2) to node[left]{$v'$} (b2);
\draw[->,dashed] (a3) to node[right]{$\mu'$} (b3);
\draw[->,dashed] (a3) to node[pos=.3,above]{$\delta$} (c1);
\draw[->] (b2) -- (c1);
\node at ([shift={(-5.5cm,-3.6cm)}]current bounding box.north east) {$\delta$};
\end{tikzpicture}
\end{center} 
 \end{document}