Exercise 8, § 7

Let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ and $(\mathrm{F}_\alpha, g_{\beta\alpha})$ be two direct systems of sets, both indexed by the same directed set I. For each $\alpha \in \mathrm{I}$, let $u_\alpha$ be a mapping of $\mathrm{E}_\alpha$ into $\mathrm{F}_\alpha$, such that the $u_\alpha$ form a direct system of mappings. Let $\mathrm{G}_\alpha \subset \mathrm{E}_\alpha \times \mathrm{F}_\alpha$ be the graph of $u_\alpha$. Show that $(\mathrm{G}_\alpha)$ is a direct system of subsets of $\mathrm{E}_\alpha \times \mathrm{F}_\alpha$ and that its direct limit may be canonically identified with the graph of $u = \varinjlim u_\alpha$.