This diagram is too complex for me to write it in full. You can start with the following TikZ code.
Code:\usetikzlibrary{positioning,arrows} \begin{tikzpicture}[auto,>=stealth'] \node (N1) {$\llap{$\tilde e:{}$}N_1 \times N_r$}; \node[right=of N1] (L1) {$\tilde L$}; \node[above=of N1] (N) {$\llap{$e_r:{}$}N_r$}; \node[above=of L1] (L) {$L_r$}; \draw[->>] (N1) -- node[swap] {$\sigma_r$} (N); \draw[->>] (L1) -- node[swap] {$\tau_r$} (L); \draw[to reversed->] (N) -- node {$\chi_r$} (L); \draw[to reversed->] (N1) -- (L1); \path[use as bounding box] (-1.3,0) rectangle (0,0); % adds to the bounding box to include \tilde e: from node N1 \end{tikzpicture}
Code:\usetikzlibrary{arrows,positioning} \begin{tikzpicture}[>=stealth',auto] \node (top) {$\llap{$\tilde e:{}$}N_1\times N_2$}; \node[below=1.5cm of top] (mid) {$\llap{$e_1\times A:{}$}N_1\times A$}; \node[below=1.5cm of mid] (bot) {}; \draw[stealth' reversed->] (top) -- node {$\sigma$} (mid); \draw[->>] (mid) -- node {$\mathsf{nat}'$} (bot); \path (top) edge[->,dashed,bend right] (bot); %or: \draw[->,dashed] (top) to[bend right] (bot); \path[use as bounding box] (-2,0) -- (0,0); \end{tikzpicture}