Remark 4.21.6. Note that a finite directed set $(I, \geq )$ always has a greatest object $i_\infty $. Hence any colimit of a system $(M_ i, f_{ii'})$ over such a set is trivial in the sense that the colimit equals $M_{i_\infty }$. In contrast, a colimit indexed by a finite filtered category need not be trivial. For instance, let $\mathcal{I}$ be the category with a single object $i$ and a single non-trivial morphism $e$ satisfying $e = e \circ e$. The colimit of a diagram $M : \mathcal{I} \to Sets$ is the image of the idempotent $M(e)$. This illustrates that something like the trick of passing to $\mathcal{I}\times \omega $ in the proof of Lemma 4.21.5 is essential.

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