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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