\documentclass[a4paper,12pt]{article}
\usepackage{amsthm,amssymb,amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix,positioning,arrows}
\usetikzlibrary{calc}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\cone}{cone}
\begin{document}
\begin{tikzpicture}
\matrix[matrix of math nodes,column sep=1cm,row sep=1cm] (m) {\Hom_R (Q^0 \oplus P^1, F)&\Hom_R (N \oplus P^0, F)&\Hom_R (M, F)&0\\
\Hom_R (Q^0 \oplus P^1, F)&\Hom_R (N, F) \oplus \Hom_R (P^0, F)&\Hom_R (M, F)&0\\ };
\draw[->] (m-1-1) -- (m-1-2);
\draw[->] (m-1-2) -- (m-1-3);
\draw[->] (m-1-3) -- (m-1-4) node[above,midway] {$\scriptstyle\alpha$};
\draw[-,double,double distance=4pt] (m-1-1) to node [left]{$\cong$} (m-2-1);
\draw[->] (m-1-2) -- (m-2-2) node[left,midway]{$\cong$};
\draw[->] (m-2-1) -- (m-2-2);
\draw[->] (m-2-2) -- (m-2-3);
\draw[->] (m-1-3) -- (m-2-3) node[left,midway]{$\cong$};
\draw[->] (m-2-3) -- (m-2-4); 

\end{tikzpicture}

\end{document}