Kerodon

Definition 9.2.1.13 ($\operatorname{Ind}$-Extensions of Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Fix regular cardinals $\kappa \leq \lambda $ and a functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Suppose we are given a functor of $\infty $-categories $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ admits $\lambda $-small $\kappa $-filtered colimits. Then there is a $(\kappa ,\lambda )$-finitary functor $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and an isomorphism of functors $\alpha : f \rightarrow F \circ H$. Moreover, the pair $(F, \alpha )$ is unique up to (canonical) isomorphism. In this case, we say that $\alpha $ exhibits $F$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$. In the special case $\kappa < \lambda = \operatorname{\textnormal{\cjRL {t}}}$, we also say that $\alpha $ exhibits $F$ as an $\operatorname{Ind}_{\kappa }$-extension of $f$. If $\kappa = \aleph _0$ and $\lambda = \operatorname{\textnormal{\cjRL {t}}}$, we say that $\alpha $ exhibits $F$ as an an $\operatorname{Ind}$-extension of $f$.