Exercise 9, § 7

Let I be an arbitrary preordered set, and let $(\mathrm{E}_\alpha)_{\alpha \in \mathrm{I}}$ be a family of sets indexed by I. For each pair of indices $(\alpha, \beta)$ such that $\alpha \leqslant \beta$, let $f_{\beta\alpha}$ be a mapping of $\mathrm{E}_\alpha$ into $\mathrm{E}_\beta$, and suppose that these mappings satisfy conditions $(\mathrm{LI_I})$ and $(\mathrm{LI_{II}})$. Let G be the set which is the sum of the family $(\mathrm{E}_\alpha)$ and (with the notation of no. 5) let $\mathrm{R}\{x, y\}$ be the relation "$\lambda(x) = \alpha \leqslant \lambda(y) = \beta$ and $y = f_{\beta\alpha}(x)$" between two elements $x$, $y$ of G. Let $\mathrm{R}'$ be the equivalence relation on G whose graph is the smallest of the graphs of equivalence relations which contain the graph of R (Chapter II, § 6, Exercise 10). The set $\mathrm{E} = \mathrm{G}/\mathrm{R}'$ is called the direct limit of the family $(\mathrm{E}_\alpha)$ with respect to the family of mappings $(f_{\beta\alpha})$, and we write $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$. When the index set I is directed, show that this definition agrees with that given in no. 5. In the general case, the restriction to $\mathrm{E}_\alpha$ of the canonical mapping of G into E is called the canonical mapping of $\mathrm{E}_\alpha$ into E and is denoted by $f_\alpha$. Suppose we are given, for each $\alpha \in \mathrm{I}$, a mapping $u_\alpha$ of $\mathrm{E}_\alpha$ into F such that $u_\beta \circ f_{\beta\alpha} = u_\alpha$ whenever $\alpha \leqslant \beta$; show that there exists a unique mapping $u$ of E into F such that $u = u_\alpha \circ f_\alpha$ for each $\alpha \in \mathrm{I}$.