Kerodon

Proof. Using Proposition 8.4.5.3, we can choose a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$. Moreover, $h$ is fully faithful, so the functor $H = \operatorname{Ind}_{\kappa }^{\lambda }(h): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is also fully faithful (Corollary 9.3.2.2). If $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete, then the functor $h$ admits a left adjoint (Proposition 8.4.5.13). It follows that $H$ also admits a left adjoint (Exercise 9.3.3.11), and therefore induces an equivalence from $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ to a reflective localization of $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$. By virtue of Corollary 7.1.4.29, it will suffice to show that the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is $\lambda $-cocomplete, which follows from Proposition 9.5.5.5. $\square$