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

\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\cone}{cone}
\begin{document}
\begin{center} {\tiny
\tikzset{node distance=3.5cm, auto}
\begin{tikzpicture}[line width=.5pt]%[style={very thick}]
  \node (a1) {$\Hom_R \big(\cone (f), F\big)$ };
    \node (a2) [right of=a1] {$\Hom_R (Q^0 \oplus P^1, F)$};
    \node (a3)[right of=a2]{$\Hom_R (N \oplus P^0, F)$};
        \node (a4)[right of=a3] {$\Hom_R (M, F)$ };
            \node (a5)[right of=a4] {$0$ };
              \node (b1)[below of=a2] {$ \Hom_R\! (Q^0\!,\! F)\! \oplus\! \Hom_R\! (P^1\!,\! F)$ };
              \node (b2)[right of=b1] {$\Hom_R (N, F) \oplus \Hom_R (P^0, F)$ };
    \node (b3) [right of=b2] {$\Hom_R (M, F)$};
        \node (b4) [right of=b3] {$0$};
            \draw[-, dashed] (a1) -- (a2); 
\draw[->] (a2) -- (a3); 
\draw[->] (a3) -- (a4); 
\draw[->] (a4) -- (a5); 
\draw[->] (b1) -- (b2); 
\draw[->] (b2) -- (b3); 
\draw[->] (b3) -- (b4); 
\draw[-,double] (a2) to node[left]{$\cong$} (b1); 
\draw[->] (a3) to node[left]{$\cong$} (b2); 
\draw[->] (a4) to node[left]{$\cong$} (b3); 
\end{tikzpicture} }
\end{center}
\end{document}