Kerodon

Proof. In what follows, we will abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with full subcategories of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$. By virtue of Proposition 9.2.3.3, it will suffice to show that every object $X \in \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ can be realized as the colimit of a $\mu $-small $\lambda $-filtered diagram in $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Using Corollary 9.2.4.21, we can realize $X$ as the colimit of a diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{K}}$ is a $\mu $-small $\kappa $-filtered $\infty $-category. If $\kappa = \lambda $, then $\operatorname{\mathcal{K}}$ is $\lambda $-filtered and there is nothing to prove. Otherwise, we can use Proposition 9.1.7.20 to realize $\operatorname{\mathcal{K}}$ as the colimit of a diagram

where $A$ is a $\mu $-small $\lambda $-directed partially ordered set and each $\operatorname{\mathcal{K}}_{\alpha }$ is a $\lambda $-small $\kappa $-filtered $\infty $-category. For each $\alpha \in A$, let $X_{\alpha }$ be a colimit of the diagram $F|_{ \operatorname{\mathcal{K}}_{\alpha } }$. Since $\operatorname{\mathcal{K}}$ is also a categorical colimit of the diagram $\{ \operatorname{\mathcal{K}}_{\alpha } \} _{\alpha \in A}$ (Proposition 9.1.6.1), we can promote the construction $\alpha \mapsto X_{\alpha }$ to a diagram $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ having colimit $X$ (see Proposition 7.5.8.12). $\square$