Exercise 6, § 7

Let I be a directed set, and let $(J_\lambda)_{\lambda \in L}$ be a family of subsets of I satisfying the conditions of Exercise 1. Let $(E_\alpha, f_{\beta\alpha})$ be a direct system of sets indexed by I, let $E = \varinjlim E_\alpha$, and for each $\lambda \in L$ let $F_\lambda$ be the direct limit of the direct system obtained from $(E_\alpha, f_{\beta\alpha})$ by restricting the index set to $J_\lambda$. Whenever $\lambda \leqslant \mu$, let $g_{\mu\lambda}$ be the canonical mapping of $F_\lambda$ into $F_\mu$ (no. 6). Show that $(F_\lambda, g_{\mu\lambda})$ is a direct system of sets relative to L, and define a canonical bijection of E onto $F = \varinjlim F_\lambda$.