Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.5.2.5. For every simplicial category $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}^{\circ }$ denote the underlying ordinary category of $\operatorname{\mathcal{C}}$ (Example 2.4.1.4). If $X$ is an object of $\operatorname{\mathcal{C}}$, then we have canonical isomorphisms

\[ (\operatorname{\mathcal{C}}_{/X})^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })_{/X} \quad \quad ( \operatorname{\mathcal{C}}_{X/})^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })_{X/}, \]

where the left hand sides are defined using the slice and coslice operations on simplicial categories (Construction 5.5.2.1 and Variant 5.5.2.3) and the right hand sides are defined using the slice and coslice operations on ordinary categories (Construction 4.3.1.1 and Variant 4.3.1.4). In other words, the slice and coslice constructions are compatible with the forgetful functor from simplicial categories to ordinary categories. We can summarize the situation more informally as follows: if $\operatorname{\mathcal{C}}$ is a category and $X$ is an object of $\operatorname{\mathcal{C}}$, then any simplicial enrichment of $\operatorname{\mathcal{C}}$ determines a simplicial enrichment on the slice and coslice categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$.