Kerodon

Remark 9.1.9.8. Following the convention of Remark 4.7.0.5, a cardinal $\kappa $ is small if it satisfies $\kappa < \operatorname{\textnormal{\cjRL {t}}}$, where $\operatorname{\textnormal{\cjRL {t}}}$ is some fixed strongly inaccessible cardinal. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-finitary (in the sense of Definition 9.1.9.3) if and only if it is $(\kappa ,\operatorname{\textnormal{\cjRL {t}}})$-finitary (in the sense of Definition 9.1.9.6). Note that in this case we automatically have $\kappa \triangleleft \operatorname{\textnormal{\cjRL {t}}}$ (Example 9.1.7.11). In particular, $F$ is finitary if and only if it is $(\aleph _0, \operatorname{\textnormal{\cjRL {t}}})$-finitary.