Remark 10.2.3.20 (Uniqueness). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. If there exists a morphism of simplicial objects $u: Y_{\bullet } \rightarrow X_{\bullet }$ which exhibits $Y_{\bullet }$ as an $n$-skeleton of $X_{\bullet }$, then $Y_{\bullet }$ is uniquely determined up to isomorphism and depends functorially on $X_{\bullet }$. To emphasize this dependence, we will denote the object $Y_{\bullet }$ by $\operatorname{sk}_{n}(X)$ and refer to to it as the $n$-skeleton of $X_{\bullet }$. By virtue of Example 10.2.3.19, this reduces to the standard definition in the special case where $\operatorname{\mathcal{C}}$ is (the nerve of) the category of sets.
Using Corollary 7.3.6.13, we see that the $n$-skeleton of a simplicial object $X_{\bullet }$ is characterized by the following universal mapping property:
If $Z_{\bullet }$ is any $n$-skeletal simplicial object of $\operatorname{\mathcal{C}}$, then composition with $u$ induces a homotopy equivalence of mapping spaces
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }})^{\operatorname{op}}, \operatorname{\mathcal{C}}) }( Z_{\bullet }, \operatorname{sk}_{n}( X_{\bullet } ) ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }})^{\operatorname{op}}, \operatorname{\mathcal{C}}) }( Z_{\bullet }, X_{\bullet } ). \]