Remark 4.8.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be a positive integer, and let $A$ be a simplicial set. Stated more informally, Proposition 4.8.4.11 asserts that $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} )$ can be viewed as a subquotient of the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(A), \operatorname{\mathcal{C}})$:
A diagram $u: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ determines a morphism from $A$ to $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ if and only if $u$ can be extended to the $(n+1)$-skeleton of $A$.
Two diagrams $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ determine the same morphism $A$ to $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ if and only if they are isomorphic relative to the $(n-1)$-skeleton of $A$.
Compare with Remark 3.5.6.13.