Kerodon

Proof. Fix an uncountable cardinal $\mu $ such that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is locally $\mu $-small. Then, for each index $i \in I$, we can choose a functor $\mathscr {F}_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and a vertex $\eta _ i \in \mathscr {F}_ i( C_ i )$ which exhibits the functor $\mathscr {F}_ i$ as corepresented by the object $C_ i$ (see Definition 5.6.6.1). Enlarging $\mu $ if necessary, we may assume that it has cofinality $\geq \lambda $ and exponential cofinality larger than the cardinality of $I$, so that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ admits $\lambda $-small colimits and $I$-indexed products (see Corollary 7.4.3.10, Variant 7.4.1.15 and Proposition 7.1.8.2). We can therefore choose a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ and a collection of natural transformations $u_ i: \mathscr {F} \rightarrow \mathscr {F}_ i |_{\operatorname{\mathcal{C}}}$ which exhibit $\mathscr {F}$ as a product of the collection $\{ \mathscr {F}_ i|_{\operatorname{\mathcal{C}}} \} _{i \in I}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. It follows that the natural transformations $u_ i$ induce a homotopy equivalence of Kan complexes $\mathscr {F}(C) \rightarrow \prod _{i \in I} \mathscr {F}_ i(C_ i)$ (see Example 7.6.1.19). We may therefore assume without loss of generality that the collection $\{ \eta _ i \} _{i \in I}$ is the image of some vertex $\eta \in \mathscr {F}(C)$. Using Remark 4.6.1.9, we see that $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $C$.

Assume first that $C$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$ and let $i$ be an element of $I$; we wish to show that $C_{i}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{i}$. Let $\delta : \operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{C}}$ be the functor given by the identity map $\operatorname{id}: \operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{C}}_ i$ together with the constant functors $\operatorname{\mathcal{C}}_ i \rightarrow \{ C_ j \} \hookrightarrow \operatorname{\mathcal{C}}_{j}$ for $j \neq i$. Our assumption that $C$ is $(\kappa ,\lambda )$-compact guarantees that the composite functor

is $(\kappa ,\lambda )$-finitary. This composite functor can be identified with the product of $\mathscr {F}_{i}$ with the constant functor $\operatorname{\mathcal{C}}_{i} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ taking the value $X = \prod _{j \neq i} \mathscr {F}_ j(C_ j)$. Since the Kan complex $X$ is nonempty (it contains the vertex $\{ \eta _ j \} _{j \neq i}$), it follows that $\mathscr {F}_ i$ is a retract of $\delta \circ \mathscr {F}$ and is therefore also $(\kappa ,\lambda )$-finitary (Remark 9.2.2.13).

We now prove the converse. Assume that each $C_ i$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{C}}_{i}$. Then each of the functors $\mathscr {F}_ i$ is $(\kappa ,\lambda )$-finitary, and therefore restricts to a $(\kappa ,\lambda )$-finitary functor $\mathscr {F}_{i}|_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ (Proposition 9.1.9.8). If $I$ is $\kappa $-small, then the formation of $I$-indexed products commutes with $\lambda $-small $\kappa $-filtered colimits in the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$ (Theorem 9.1.5.9), so the functor $\mathscr {F} = \prod _{i \in I} \mathscr {F}_ i|_{\operatorname{\mathcal{C}}}$ is also $(\kappa ,\lambda )$-finitary: that is, the object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. $\square$