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Proposition 3.5.3.17 (Uniqueness). Let $n$ be an integer and let $f: X \rightarrow Y$ be a morphism of simplicial sets, where $Y$ is $n$-coskeletal. The following conditions are equivalent:

$(1)$

The morphism $f$ exhibits $Y$ as an $n$-coskeleton of $X$: that is, it is bijective on $m$-simplices for $m \leq n$.

$(2)$

For every $n$-coskeletal simplicial set $Z$, composition with $f$ induces an isomorphism of simplicial sets $\operatorname{Fun}( Y, Z ) \rightarrow \operatorname{Fun}(X, Z)$.

$(3)$

For every $n$-coskeletal simplicial set $Z$, composition with $f$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(Y, Z) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(X, Z). \]

Proof. Assertion $(2)$ is equivalent to the requirement that, for every $n$-coskeletal simplicial set $Z$ and every simplicial set $K$, composition with $f$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(Y,Z ) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(X,Z) ). \]

By virtue of Corollary 3.5.3.12, we can replace $Z$ by $\operatorname{Fun}(K, Z)$ and thereby reduce to the case $K = \Delta ^0$. This proves the equivalence $(2) \Leftrightarrow (3)$.

The implication $(1) \Rightarrow (3)$ follows immediately from Proposition 3.5.3.10. We will complete the proof by showing that $(3)$ implies $(1)$. Using Proposition 3.5.3.16, we can choose a morphism $u: X \rightarrow \operatorname{cosk}_{n}(X)$ which exhibits $\operatorname{cosk}_{n}(X)$ as an $n$-coskeleton of $X$. Then $u$ satisfies condition $(3)$, so $f$ factors (uniquely) as a composition $X \xrightarrow {u} \operatorname{cosk}_{n}(X) \xrightarrow {g} Y$. We can therefore replace $X$ by $\operatorname{cosk}_{n}(X)$ and thereby reduce to the case where $X$ is $n$-coskeletal. In this case, condition $(3)$ implies that $f$ is an isomorphism of ($n$-coskeletal) simplicial sets, and therefore bijective on $m$-simplices for $m \leq n$. $\square$