\documentclass{standalone} \usepackage{tikz} \begin{document} \begin{tikzpicture}[auto]

\node (a) at (1, 1) {$1$}; \node (b) at (1,0) {$1 \times 1$}; \node (c) at (1, -1) {$X \times A$}; \node (d) at (3, -1) {$B$}; \node (e) at (2,0) {$A$}; \draw[->] (a) to node[swap] {$\scriptstyle \delta_1$} (b); \draw[->] (b) to node[swap] {$\scriptstyle x \times a$} (c); \draw[->] (e) to node {$\scriptstyle g$} (d); \draw[->] (a) to node {$\scriptstyle a$} (e); \draw[->] (c) to node[swap] {$\scriptstyle \tilde{f}$} (d);

\node (f) at (5, -1) {$B^A \times A$}; \node (g) at (5,1) {$X \times A$}; \node (h) at (7, -1) {$B$}; \draw[->] (g) to node[swap] {$\scriptstyle f \times \mathrm{id}_{A}$} (f); \draw[->] (g) to node {$\scriptstyle \tilde{f}$} (h); \draw[->] (f) to node[swap] {$\scriptstyle \epsilon$} (h); \end{tikzpicture} \end{document}