Notation 3.5.3.18. Let $X$ be a simplicial set and let $n$ be an integer. It follows from Proposition 3.5.3.16 that there exists a morphism of simplicial sets $f: X \rightarrow Y$ which exhibits $Y$ as an $n$-coskeleton of $X$. Moreover, Proposition 3.5.3.17 guarantees that $Y$ is unique up to (canonical) isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{cosk}_{n}(X)$ and refer to it as the $n$-coskeleton of $X$. More explicitly, we can take $\operatorname{cosk}_{n}(X)$ to be the simplicial set constructed in the proof of Proposition 3.5.3.16, given by the construction
\[ ( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ). \]