Lemma 4.22.11. Let $\mathcal{C}$ be a category. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of filtered index categories. If $H$ is cofinal, then any diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ H$ is essentially constant.

Proof. This follows formally from Lemmas 4.22.9 and 4.17.2. $\square$

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