Remark 3.5.6.13. Let $X$ be a Kan complex, let $n \geq 0$ be an integer, and let $A$ be a simplicial set. Stated more informally, Proposition 3.5.6.12 asserts that $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \pi _{\leq n}(X) )$ can be viewed as a subquotient of the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(A), X)$:
A morphism $u: \operatorname{sk}_{n}(A) \rightarrow X$ determines a map from $A$ to $\pi _{\leq n}(X)$ if and only if $u$ can be extended to the $(n+1)$-skeleton of $A$.
Two such morphisms $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow X$ determine the same map from $A$ to $\pi _{\leq n}(X)$ if and only if they are homotopic relative to the $(n-1)$-skeleton of $A$.