Corollary 2

Let $(\mathrm{E}^\lambda_\alpha, f^\lambda_{\alpha\beta})_{\lambda \in \mathrm{L}}$ be a family of inverse systems of sets relative to I. If $\prod_{\lambda \in \mathrm{L}} f^\lambda_{\alpha\beta}$ denotes the extension to products (Chapter II, § 5, no. 7, Definition 2) of the family of mappings $(f^\lambda_{\alpha\beta})_{\lambda \in \mathrm{L}}$, then $\left( \prod_{\lambda \in \mathrm{L}} \mathrm{E}^\lambda_\alpha, \prod_{\lambda \in \mathrm{L}} f^\lambda_{\alpha\beta} \right)$ is an inverse system of sets relative to I, and (up to a canonical bijection) we have

Consider the double inverse system $(\mathrm{E}^\lambda_\alpha, g^{\lambda\mu}_{\alpha\beta})$ relative to $\mathrm{I} \times \mathrm{L}$, where the order relation on L is equality (no. 1, Example 1), and apply Proposition 4.