Corollary 9.1.8.9. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category having countably many simplices. Then there exists a right cofinal functor $F: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$. Here $\operatorname{\mathbf{Z}}_{\geq 0}$ denotes the set of non-negative integers, equipped with its usual ordering.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Theorem 9.1.8.7, we can reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(A)$, where $(A, \leq )$ is a countable directed partially ordered set. In this case, we can identify a functor $F: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ with a nondecreasing sequence $a_0 \leq a_1 \leq a_2 \leq \cdots $. To guarantee that $F$ is right cofinal, it suffices to choose a sequence with the property that each $a \in A$ satisfies $a \leq a_ n$ for $n \gg 0$ (see Theorem 7.2.3.1). $\square$