$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Broadly speaking, Proposition will be useful to us because it shows that the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$ admits a (left and right) cofinal diagram $Q: K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where the simplicial set $K$ is finite-dimensional. Beware that it is not possible to arrange that the simplicial set $K$ is finite, since an $\infty $-category which admits finite colimits need not be idempotent complete (Warning In particular, there does not exist a categorical equivalence $K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where $K$ is a finite simplicial set.