Kerodon

Remark 5.5.2.12. 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 $\underline{\operatorname{\mathcal{C}}}^{\triangleleft }$ and $\underline{\operatorname{\mathcal{C}}}^{\triangleright }$ of Notation 5.5.2.8 and Variant 5.5.2.9 are also constant, associated to the ordinary categories $\operatorname{\mathcal{C}}^{\triangleleft }$ and $\operatorname{\mathcal{C}}^{\triangleright }$ of Example 4.3.2.5. In other words, the formation of cones is compatible with the operation of regarding an ordinary category as a (constant) simplicial category.