Kerodon

Proof. We first prove $(1)$. We proceed as in the proof of Proposition 3.5.3.23. Fix integers $0 < i < m$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{cosk}_ n(\operatorname{\mathcal{C}})$. Using Remark 3.5.3.21, we can identify $\sigma _0$ with a morphism of simplicial sets $f_0: \operatorname{sk}_{n}( \Lambda ^{m}_{i} ) \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $f_0$ can be extended to the $n$-skeleton of $\Delta ^{m}$. If $n < m-1$, then $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n}( \Delta ^{m} )$ and there is nothing to prove. We may therefore assume that $n \geq m-1$, so that $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \Lambda ^{m}_{i}$. In this case, our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that $f_0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.

We now prove $(2)$. By construction, the tautological map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is bijective on objects, and therefore essentially surjective. By virtue of Proposition 4.6.5.10, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of pinched morphism spaces

exhibits $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) }(X,Y)$ as an $(n-2)$-truncation of $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y )$. This is a special case of Example 3.5.7.23, since $\theta $ exhibits $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) }(X,Y)$ as an $(n-1)$-coskeleton of $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y )$ (Remark 4.6.5.4). $\square$