# Kerodon

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

Corollary 4.3.2.17.

• For any category $\operatorname{\mathcal{D}}$, the join functor

$\operatorname{Cat}\rightarrow \operatorname{Cat}_{\operatorname{\mathcal{D}}/ } \quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$

admits a right adjoint, given on objects by the slice construction $(G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}) \mapsto \operatorname{\mathcal{E}}_{/G}$.

• For any category $\operatorname{\mathcal{C}}$, the join functor

$\operatorname{Cat}\rightarrow \operatorname{Cat}_{\operatorname{\mathcal{C}}/ } \quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$

admits a right adjoint, given on objects by the coslice construction $(F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}) \mapsto \operatorname{\mathcal{E}}_{F/}$.