Corollary

Tag 03OQ, permanent. Cite it as [Bourbaki, Tag 03OQ].

Theory of Sets, chapter III, § 7. Inverse limits and direct limits

Let $(\mathrm{{E}'}^\lambda_\alpha, {f'}^{\mu\lambda}_{\beta\alpha})$ be another direct system of sets relative to $\mathrm{I} \times \mathrm{L}$ , and for each $(\alpha, \lambda) \in \mathrm{I} \times \mathrm{L}$ let $u_\alpha^\lambda$ be a mapping of $\mathrm{E}_\alpha^\lambda$ into $\mathrm{{E}'}^\lambda_\alpha$ , such that the $u_\alpha^\lambda$ form a direct system of mappings. Then we have

(33)

\varinjlim_{\alpha,\ \lambda} u_\alpha^\lambda = \varinjlim_{\lambda}\ (\varinjlim_{\alpha} u_\alpha^\lambda).

We leave the verification to the reader.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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