# Kerodon

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

Warning 4.3.2.8. The join operation of Definition 4.3.2.1 is not commutative. For example, if $\operatorname{\mathcal{C}}$ is a category, then the left cone $\operatorname{\mathcal{C}}^{\triangleleft }$ need not be isomorphic (or even equivalent) to the right cone $\operatorname{\mathcal{C}}^{\triangleright }$. However, we do have canonical isomorphisms

$(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq \operatorname{\mathcal{D}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}^{\operatorname{op}},$

depending functorially on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.