Example 9.4.2.12. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, suppose we are given a $\kappa $-small collection of functors $\{ F_ i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$, where each of the $\infty $-categories $\operatorname{\mathcal{D}}_{i}$ is $(\kappa ,\lambda )$-cocomplete. If the induced functor
\[ F: \operatorname{\mathcal{C}}\rightarrow \prod _{i \in I} \operatorname{\mathcal{D}}_{i} \]
is $(\kappa ,\lambda )$-compact, then each of the functors $F_ i$ is $(\kappa ,\lambda )$-compact. The converse holds if the index set $I$ is $\kappa $-small. See Proposition 9.2.8.6.