Kerodon

Proof. Assume that $F$ is accessible; we will show that condition $(2)$ is satisfied (the converse is immediate from the definitions). Using Remark 9.4.8.2, we can choose a small regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\kappa $-accessible and the functor $F$ is $\kappa $-finitary. Let $\operatorname{\mathcal{C}}_{< \kappa }$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects. It follows from Proposition 9.4.7.2 that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially small. Using Remark 9.4.7.8, we can choose a small regular cardinal $\lambda \geq \kappa $ such that $F( \operatorname{\mathcal{C}}_{< \kappa } ) \subseteq \operatorname{\mathcal{D}}_{< \lambda }$: that is, $F$ carries $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ to $\lambda $-compact objects of $\operatorname{\mathcal{D}}$. Enlarging $\lambda $ if necessary, we may assume that $\kappa \trianglelefteq \lambda $ (see Proposition 9.1.7.8). We will complete the proof by showing that the functor $F$ is $\lambda $-compact. Since the functor $F$ is $\kappa $-finitary, it is also $\lambda $-finitary. It will therefore suffice to show that for every $\lambda $-compact object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is also $\lambda $-compact. Using Corollary 9.3.6.7, we can realized $C$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram $G: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$. Since the functor $F$ is $\kappa $-finitary, the object $F(C)$ is a colimit of the composite functor

and is therefore $\lambda $-compact by virtue of Corollary 9.2.5.25. $\square$