Proposition 10

Let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ and $(\mathrm{E}'_\alpha, f'_{\beta\alpha})$ be two direct systems of sets, both relative to the same directed set I. Let $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$, $\mathrm{E}' = \varinjlim \mathrm{E}'_\alpha$, and let $f_\alpha : \mathrm{E}_\alpha \to \mathrm{E}, f'_\alpha : \mathrm{E}'_\alpha \to \mathrm{E}'$ denote the canonical mappings, for each $\alpha \in \mathrm{I}$. Then $(\mathrm{E}_\alpha \times \mathrm{E}'_\alpha, f_{\beta\alpha} \times f'_{\beta\alpha})$ is a direct systems of sets, $(f_\alpha \times f'_\alpha)$ is a direct system of mappings, and $\varinjlim (f_\alpha \times f'_\alpha)$ is a bijection

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The first two assertions of the Proposition are immediately verified. To show that $g = \varinjlim (f_\alpha \times f'_\alpha)$ is bijective, we apply Proposition 6 of no. 6. Clearly $\mathrm{E} \times \mathrm{E}'$ is the union of the sets $f_\alpha(\mathrm{E}_\alpha) \times f'_\alpha(\mathrm{E}'_\alpha)$; hence $g$ is surjective. If $(x, x')$ and $(y, y')$ are two elements of $\mathrm{E}_\alpha \times \mathrm{E}'_\alpha$ such that $f_\alpha(x) = f_\alpha(y)$ and $f'_\alpha(x') = f'_\alpha(y')$, then (no. 5, Lemma 1) there exist two elements $\beta$, $\gamma$ of I such that $\beta \geqslant \alpha$, $\gamma \geqslant \alpha$, and $f_{\beta\alpha}(x) = f_{\beta\alpha}(y)$, $f'_{\gamma\alpha}(x') = f'_{\gamma\alpha}(y')$; since I is directed, there exists $\delta \in \mathrm{I}$ such that $\delta \geqslant \beta$ and $\delta \geqslant \gamma$; hence $f_{\delta\alpha}(x) = f_{\delta\alpha}(y)$ and $f'_{\delta\alpha}(x') = f'_{\delta\alpha}(y')$. This completes the proof.

The bijection $g$ is called canonical.