Proposition 3

Let $(E_\alpha, f_{\beta \alpha}), (E'_\alpha, f'_{\beta \alpha}), (E''_\alpha, f''_{\beta \alpha})$ be three direct systems of $A_\alpha$*-modules and* $(u_\alpha), (v_\alpha)$ two direct systems of $A_\alpha$*-linear mappings such that the sequences*

are exact for all $\alpha$. Then, writing $u = \lim \to u_\alpha, v = \lim \to v_\alpha$, the sequence

is exact.

$u(\lim \to E'_\alpha) = \lim \to u_\alpha(E'_\alpha)$ and $v^{-1}(0) = \lim \to v_\alpha^{-1}(0)$ (Set Theory, III, § 7, no. 6, Corollary to Proposition 7).

Loosely speaking, Proposition 3 can also be expressed by saying that passing to the direct limit preserves exactness.