Remark 1

For each $\alpha \in I$ put $E'_\alpha = f_\alpha(E)$. Then the sets $E'_\alpha$ form an inverse system of subsets of the $E_\alpha$ by reason of (2), and it is immediately clear that $\varprojlim E'_\alpha = E = \varprojlim E_\alpha$. The mapping $f'_{\alpha\beta} : E'_\beta \to E'_\alpha$ (where $\alpha \leqslant \beta$), whose graph is the same as that of the restriction of $f_{\alpha\beta}$ to $E_\beta$, is surjective, and we have

for all $\alpha \in I$.