Proposition 10.2.4.21 (Existence of Coskeleta). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite limits and let $n$ be an integer. Then every (semi)simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ admits an $n$-coskeleton $\operatorname{cosk}_{n}(X)_{\bullet }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will prove the assertion for simplicial objects; the analogous statement for semisimplicial objects is similar (but easier). It will suffice to show that the functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n})^{\operatorname{op}} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a right Kan extension $Y_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$; Corollary 7.3.6.13 then guarantees that there is an (essentially unique) morphism of simplicial objects $u: X_{\bullet } \rightarrow Y_{\bullet }$ which is the identity when restricted to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n} )^{\operatorname{op}}$. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $k$, the diagram
\[ G: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n} )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow {X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a limit. As in the proof of Proposition 10.2.4.15, we observe that the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n} )$ is right cofinal, where $\operatorname{\mathcal{J}}\subseteq \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n}$ is the full subcategory spanned by the injective maps $[m] \hookrightarrow [k]$. We are therefore reduced to showing that $G|_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})^{\operatorname{op}} }$ has a limit in $\operatorname{\mathcal{C}}$ (Corollary 7.2.2.10), which follows from our assumption that $\operatorname{\mathcal{C}}$ admits finite limits (since $\operatorname{\mathcal{J}}$ is the category associated to a finite partially ordered set). $\square$