Kerodon

Proof. Fix an integer $k > n$, and let $\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} \rightarrow \operatorname{\mathcal{C}}$ be the $k$th degeneracy cube of the simplicial object $X_{\bullet }$. Set $K = \{ 1, 2, \cdots , k \} $, and let $P^{> 0}(K)$ denote the collection of all nonempty subsets of $K$. Then $\sigma _{k}$ can be identified with a functor $\sigma _{k}^{\circ }$ from $\operatorname{N}_{\bullet }( P^{> 0}(K) )$ to the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ X_0 / }$. Our assumption that $X_{\bullet }$ is $n$-skeletal guarantees that $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.2.3.14), or equivalently that $\sigma _{k}^{\circ }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{X_0/}$ (Remark 7.1.3.11). Since the functor $F$ preserves pushouts, the induced functor of coslice $\infty $-categories $F_{X_0/}: \operatorname{\mathcal{C}}_{X_0/} \rightarrow \operatorname{\mathcal{D}}_{ F(X_0)/ }$ preserves finite colimits (Example 7.6.2.31). In particular, $F_{X_0/} \circ \sigma _{k}^{\circ }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}_{ F(X_0) / }$, so that $F \circ \sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Remark 7.1.3.11). Allowing $k$ to vary, we conclude that $F( X_{\bullet } )$ is an $n$-skeletal simplicial object of $\operatorname{\mathcal{D}}$ (Proposition 10.2.3.14). $\square$