Remark 2

Let I be a (right) directed ordered set, let $(E_\alpha, f_{\alpha\beta})$ be an inverse system of sets relative to I, and for each $\alpha \in I$ let $u_\alpha : F \to E_\alpha$ be a mapping such that the family $(u_\alpha)$ satisfies formula (5). Consider the inverse system $(F_\alpha, i_{\alpha\beta})$ indexed by I, where $F_\alpha = F$ for all $\alpha \in I$ and $i_{\alpha\beta}$ is the identity mapping of F. Then (no. 1, Example 2) F is canonically identified with $\varprojlim F_\alpha$. If we consider $u_\alpha$ as a mapping of $F_\alpha$ into $E_\alpha$, then $(u_\alpha)$ is an inverse system of mappings, and the mapping $u : F \to E$ defined by (6) is identified with the inverse limit of this system of mappings. Hence, by abuse of language, we write $u = \varprojlim u_\alpha$.