Remark 4.6.5.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $n \geq 0$ be an integer, and let $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ denote the $n$-coskeleton of $\operatorname{\mathcal{C}}$ (Notation 3.5.3.18). For every pair of vertices $X,Y \in \operatorname{\mathcal{C}}$, Remark 4.3.5.16 supplies canonical isomorphisms
\[ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})_{X/} \simeq \operatorname{cosk}_{n-1}( \operatorname{\mathcal{C}}_{X/} ) \times _{ \operatorname{cosk}_{n-1}(\operatorname{\mathcal{C}}) } \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}) \]
\[ \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})_{/Y} \simeq \operatorname{cosk}_{n-1}( \operatorname{\mathcal{C}}_{/Y} ) \times _{ \operatorname{cosk}_{n-1}(\operatorname{\mathcal{C}}) } \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}). \]
Passing to fibers over the vertices $Y$ and $X$, we obtain isomorphisms of pinched morphism spaces
\[ \operatorname{Hom}_{\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y) \simeq \operatorname{cosk}_{n-1}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) ) \quad \operatorname{Hom}_{\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})}^{\mathrm{R}}(X,Y) \simeq \operatorname{cosk}_{n-1}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) ). \]
In particular, if $\operatorname{\mathcal{C}}$ is $n$-coskeletal, then the pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ are $(n-1)$-coskeletal.