Kerodon

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

Corollary 7.2.1.12. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there exists a partially ordered set $(A, \leq )$ and a morphism of simplicial sets $F: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$ which is both left and right cofinal. Moreover, if the simplicial set $\operatorname{\mathcal{C}}$ is finite, then we can arrange that the partially ordered set $(A, \leq )$ is finite.

Proof. Combine Theorem 6.3.7.1 (and Variant 6.3.7.17) with Corollary 7.2.1.11. $\square$