Is the skeleton-coskeleton adjunction $sSet$-enriched? Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?

Well, first we would have to have simplicially enriched functors. But $\mathrm{sk}_n : \mathbf{sSet} \to \mathbf{sSet}$ is _not_ simplicially enriched. Indeed, any simplicially enriched functor must preserve simplicial homotopy equivalences, but $\mathrm{sk}_n$ does _not_ preserve simplicial homotopy equivalence. (Consider the unique morphism $\Delta^{n+1} \to \Delta^0$: this is a simplicial homotopy equivalence, but $\operatorname{sk}_n \Delta^{n+1} \to \operatorname{sk}_n \Delta^n$ is _never_ a simplicial homotopy equivalence.)

On the other hand, $\mathrm{cosk}_n : \mathbf{sSet} \to \mathbf{sSet}$ admits a unique simplicial enrichment making the canonical natural transformation $\mathrm{id} \Rightarrow \mathrm{cosk}_n$ simplicially enriched.