Kerodon

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

Remark 5.4.2.11. 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). Then we have canonical isomorphisms

\[ ( \operatorname{\mathcal{C}}^{\triangleleft } )^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })^{\triangleleft } \quad \quad ( \operatorname{\mathcal{C}}^{\triangleright } )^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })^{\triangleright }, \]

where the left hand sides are defined using Notation 5.4.2.8 and Variant 5.4.2.9, and the right hand sides are defined in Example 4.3.2.5. In other words, the formation of cones is compatible with the forgetful functor from simplicial categories to ordinary categories.