Example 4.3.3.27. Let $\operatorname{\mathcal{C}}$ be a category. Then Example 4.3.3.14 supplies canonical isomorphisms
\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\triangleleft } \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\triangleleft } ) \quad \quad \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\triangleright } \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\triangleright } ), \]
where $\operatorname{\mathcal{C}}^{\triangleleft }$ and $\operatorname{\mathcal{C}}^{\triangleright }$ denote the left and right cones of $\operatorname{\mathcal{C}}$ (see Example 4.3.2.5).