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In Def 1.1 this paper it says about a functor between triangulated categories $i:C \rightarrow D$ is cofinal.

This does not seem to be the usual one for categories. Am I correct in understanding that it means

$i$ is cofinal iff For all $d \in D$ exists some $c$, such that $d$ is a direct some of $i(c)$?

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As the definition says, $i$ must be fully faithful and every object of $D$ must be a summand of an object in the image of $i$. You are right that this is essentially unrelated to the generally category theory notion of cofinality.

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