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Remark In ยง, we will give a different description of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ (at least up to equivalence): it can be realized as the $\infty $-category of pointed objects of the $\infty $-category $\operatorname{\mathcal{S}}$ (that is, the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}})$ spanned by those diagrams $f: X \rightarrow Y$ where Kan complex $X$ is contractible; see Example ). Beware that the analogous statement does not hold at the level of homotopy categories: there is no formal mechanism to extract the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ from the unpointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Warning