Example 2

Suppose that, for each $\alpha \in \mathrm{I}$, $\mathrm{E}_\alpha$ is the same set F and that whenever $\alpha \leqslant \beta$, $f_{\beta\alpha}$ is the identity mapping of F onto itself. Then there exists a canonical bijection of $\varinjlim \mathrm{E}_\alpha$ onto F. In order to define $\varinjlim \mathrm{E}_\alpha$, we have to form the set G which is the sum of the family $(\mathrm{E}_\alpha)$; G is therefore the union of a family $(\mathrm{G}_\alpha)$ of mutually disjoint sets, and for each $\alpha \in \mathrm{I}$ there is a canonical bijection $h_\alpha : \mathrm{F} \to \mathrm{G}_\alpha$. We have next to consider the equivalence relation R on G corresponding to the partition $(\mathrm{P}_y)_{y \in \mathrm{F}}$, where $\mathrm{P}_y$ is the set of all $h_\alpha(y)$ as $\alpha$ runs through I. Clearly $y \to \mathrm{P}_y$ is a bijection whose inverse is the bijection required. We shall identify F with $\varinjlim \mathrm{E}_\alpha$ by means of this canonical bijection.