Notation 9.2.1.19. Let $\kappa \leq \lambda < \mu $ be regular cardinals, where $\mu $ has exponential cofinality $\geq \lambda $. Proposition 9.2.1.18 asserts that there exists an essentially unique $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$. To emphasize its uniqueness, we will typically denote the functor $T(-)$ by $\operatorname{Ind}_{\kappa }^{\lambda }(-)$ (compare with Notation 9.2.1.8).
Following the convention of Remark 4.7.0.5, we say that $\kappa $ is small if it satisfies $\kappa < \operatorname{\textnormal{\cjRL {t}}}$, for some fixed strongly inaccessible cardinal $\operatorname{\textnormal{\cjRL {t}}}$. In this case, we typically denote the functor $\operatorname{Ind}_{\kappa }^{\operatorname{\textnormal{\cjRL {t}}}}(-)$ by $\operatorname{Ind}_{\kappa }(-)$. In the special case where $\kappa = \aleph _0$, we also denote the functor $\operatorname{Ind}_{\kappa }(-)$ by $\operatorname{Ind}(-)$.