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 $(F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}) \mapsto \operatorname{\mathcal{E}}_{F/}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$