Notation 3.5.4.19. Let $X$ be a simplicial set and let $n$ be an integer. It follows from Proposition 3.5.4.17 that there exists a morphism of simplicial sets $f: X \rightarrow Y$ which exhibits $Y$ as a weak $n$-coskeleton of $X$. Moreover, Proposition 3.5.4.18 guarantees that $Y$ is unique up to (canonical) isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{cosk}^{\circ }_{n}(X)$ and refer to it as *the* weak $n$-coskeleton of $X$. More explicitly, we can take $\operatorname{cosk}^{\circ }_{n}(X)$ to be the image of the restriction map $\operatorname{cosk}_{n+1}(X) \rightarrow \operatorname{cosk}_{n}(X)$ (see the proof of Proposition 3.5.4.17).

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