Lemma 10.8.4. Let $(M_ i, \mu _{ij})$ be a directed system. Let $M = \mathop{\mathrm{colim}}\nolimits M_ i$ with $\mu _ i : M_ i \to M$. Then, $\mu _ i(x_ i) = 0$ for $x_ i \in M_ i$ if and only if there exists $j \geq i$ such that $\mu _{ij}(x_ i) = 0$.

**Proof.**
This is clear from the description of the directed colimit in Lemma 10.8.3.
$\square$

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