Kerodon

Remark 5.5.2.6. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\underline{\operatorname{\mathcal{C}}}$ denote the associated constant simplicial category (Example 2.4.2.4). Then the simplicial categories $\underline{\operatorname{\mathcal{C}}}_{/X}$ and $\underline{\operatorname{\mathcal{C}}}_{X/}$ of Construction 5.5.2.1 and Variant 5.5.2.3 are also constant, associated to the ordinary categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ of Construction 4.3.1.1 and Variant 4.3.1.4, respectively. In other words, the slice and coslice constructions are compatible with the operation of regarding an ordinary category as a (constant) simplicial category.