Remark 9.2.3.4. We will see later that, if the functor $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$, then it satisfies the following stronger version of condition $(3)$:
- $(3^+)$
Every object $\operatorname{\mathcal{D}}$ can be realized as the colimit of a diagram
\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow {f} \operatorname{\mathcal{D}}, \]where $\operatorname{\mathcal{K}}$ is a $\lambda $-small, $\kappa $-filtered $\infty $-category.
In other words, every object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ can be built in a single step as a $\lambda $-small $\kappa $-filtered colimit of objects of $\operatorname{\mathcal{C}}$ (rather than iteratively). See Corollary 9.2.4.21.