Proof.
Condition $(3)$ is equivalent to the requirement that, for every weakly $n$-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 3.5.4.13, we can replace $Z$ by $\operatorname{Fun}(K, Z)$ and thereby reduce to the case $K = \Delta ^0$. This proves the equivalence $(2) \Leftrightarrow (3)$.
We next show that $(1)$ implies $(3)$. Assume that condition $(1)$ is satisfied, and let $Z$ be a weakly $n$-coskeletal simplicial set. We then have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, Z ) \ar [r]^-{ \circ f} \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, Z ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(Y), Z) \ar [r]^-{\circ f} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(X), Z ). } \]
Since $f$ is bijective on $m$-simplices for $m \leq n$, the lower horizontal map is a bijection. Using Proposition 3.5.4.12, we see that the vertical maps are injective. Consequently, to prove $(3)$, it will suffice to show that their images agree (under the bijection provided by the the lower horizontal map). This follows from Proposition 3.5.4.12, together with our assumption that $f$ is surjective on $(n+1)$-simplices.
We now show that $(3)$ implies $(1)$. Using Proposition 3.5.4.17, we can choose a morphism $u: X \rightarrow \operatorname{cosk}^{\circ }_{n}(X)$ which exhibits $\operatorname{cosk}^{\circ }_{n}(X)$ as a weak $n$-coskeleton of $X$. Then $u$ satisfies condition $(3)$, so $f$ factors (uniquely) as a composition $X \xrightarrow {u} \operatorname{cosk}^{\circ }_{n}(X) \xrightarrow {g} Y$. We can therefore replace $X$ by $\operatorname{cosk}^{\circ }_{n}(X)$ and thereby reduce to the case where $X$ is weakly $n$-coskeletal. Condition $(3)$ then guarantees that $f$ is an isomorphism of (weakly $n$-coskeletal) simplicial sets, so condition $(1)$ is automatic.
$\square$