Example 3.5.7.24. Let $X$ be a Kan complex and let $n$ be an integer. Then the tautological map $X \rightarrow \operatorname{cosk}^{\circ }_{n}(X)$ exhibits the weak coskeleton $\operatorname{cosk}^{\circ }_{n}(X)$ as an $n$-truncation of $X$. This follows from Example 3.5.7.23 and Remark 3.5.7.22, since the quotient map $\operatorname{cosk}_{n+1}(X) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a trivial Kan fibration (Proposition 3.5.4.22). Alternatively, it can be deduced directly from Remark 3.5.4.16.
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