Kerodon

Proof. Choose a functor $g: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits, the functor $g$ admits a left adjoint $f: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$. Then the functor $F = \operatorname{Ind}_{\lambda }^{\mu }(f)$ is left adjoint to $G = \operatorname{Ind}_{\lambda }^{\mu }(g)$ (Exercise 9.2.1.22). Since $g$ is fully faithful, the functor $G$ is also fully faithful (Corollary 9.2.3.2), and therefore induces an equivalence from $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ to a reflective subcategory of $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}} )$ (Remark 6.3.3.4). By virtue of Corollary 7.1.4.23, it will suffice to show that the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}} )$ admits $\mu $-small $\kappa $-filtered colimits. This follows immediately from the equivalence $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}}) \simeq \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ supplied by Theorem 9.2.9.4. $\square$