Lemma 10.2.3.13. Let $k \geq 0$ and $n$ be integers, let $K = \{ 1, \cdots , k \} $, and let $P^{\leq n}(K)$ denote the partially ordered collection of all subsets $J \subseteq K$ which have cardinality $\leq n$. Then the assignment $J \mapsto \alpha _{J}$ of Construction 10.2.3.10 determines a right cofinal functor of $\infty $-categories
\[ \alpha : \operatorname{N}_{\bullet }( P^{\leq n}(K) ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ [k] / }^{\leq n})^{\operatorname{op}}. \]