Kerodon

Exercise 9.2.1.22. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Show that $\operatorname{Ind}_{\kappa }^{\lambda }(G): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is right adjoint to $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$.