Remark

Suppose that, for each $\alpha \in \mathrm{I}$, $u_\alpha : \mathrm{E}_\alpha \to \mathrm{E}'$ is a mapping such that the family $(u_\alpha)$ satisfies (23). Consider the direct system $(\mathrm{E}_\alpha, i_{\beta\alpha})$ relative to I, where $\mathrm{E}'_\alpha = \mathrm{E}'$ for all $\alpha \in \mathrm{I}$, and $i_{\beta\alpha}$ is the identity mapping of $\mathrm{E}'$. Then (no. 5, Example 2) $\mathrm{E}'$ can be identified canonically with $\varinjlim \mathrm{E}'_\alpha$. If $u_\alpha$ is considered as a mapping of $\mathrm{E}_\alpha$ into $\mathrm{E}'_\alpha$, then $(u_\alpha)$ is a direct system of mappings, and the mapping $u : \mathrm{E} \to \mathrm{E}'$ defined by (24) is identified with the direct limit of this system of mappings. Hence, by abuse of language, we write $u = \varinjlim u_\alpha$.

If J is a subset of I which is directed (with respect to the induced pre-ordering), it is clear that the pair consisting of the subfamily $(\mathrm{E}_\alpha)_{\alpha \in \mathrm{J}}$ and the family $(f_{\beta\alpha})$, where $\alpha \leqslant \beta$ and $\alpha \in \mathrm{J}$ and $\beta \in \mathrm{J}$, is a direct system of sets relative to J; it is said to be obtained by restricting the index set to J. Let E, E′ respectively denote the direct limits of the families $(\mathrm{E}_\alpha)_{\alpha \in \mathrm{I}}$ and $(\mathrm{E}_\alpha)_{\alpha \in \mathrm{J}}$, and for each $\alpha \in \mathrm{I}$ let $f_\alpha : \mathrm{E}_\alpha \to \mathrm{E}$ denote the canonical mapping. Then $(f_\alpha)_{\alpha \in \mathrm{J}}$ is a direct system of mappings, and consequently $g = \varinjlim f_\alpha$ is a mapping of E′ into E, called canonical. Moreover, if J′ is a directed subset of J, if E″ is the direct limit of the family $(\mathrm{E}_\alpha)_{\alpha \in \mathrm{J}'}$, and if

are the canonical mappings, then it follows immediately from Proposition 6 that

(28)