Variant 10.2.5.14. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{+,\operatorname{inj}}^{\leq n}$ denote the category whose objects are linearly ordered sets $[m] = \{ 0 < 1 < \cdots < n\} $ for $-1 \leq m \leq n$, and whose morphisms are strictly increasing functions. We say that an augmented semisimplicial object $C_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ is $n$-coskeletal if the functor
\[ C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+, \operatorname{inj}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}} \]
is a right Kan extension of its restriction to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+, \operatorname{inj}} )^{\operatorname{op}}$.