Kerodon

Proof. We first prove $(1)$. Assume that $F$ is $\kappa $-left exact; we wish to show that it preserves $\kappa $-small limits. Equivalently, we wish to show that if $G: K \rightarrow \operatorname{\mathcal{C}}$ is a $\kappa $-small diagram, then the functor $F_{/G}: \operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{D}}_{ /(F \circ G)}$ preserves final objects. It follows from Corollary 9.2.5.21 that the functor $F_{/G}$ is also $\kappa $-left exact. In particular, it is right cofinal (Corollary 9.2.5.17), and therefore preserves final objects by virtue of Corollary 7.2.1.9.

We now prove $(2)$. Assume that $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits and that $F$ preserves $\kappa $-small limits; we wish to show that $F$ is $\kappa $-left exact. By virtue of Theorem 9.2.5.15, it will suffice to show that for every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered. Fix a $\kappa $-small simplicial set $K$ and a diagram $\widetilde{G}: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}$; we wish to show that $\widetilde{G}$ can be extended to the cone $K^{\triangleleft }$. Let us identify $\widetilde{G}$ with a pair $(G, D')$, where $G: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram in $\operatorname{\mathcal{C}}$ and $D'$ is an object of the $\infty $-category $\operatorname{\mathcal{D}}_{/ (F \circ G)}$ lying over $D$. Since $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, we can extend $G$ to a limit diagram $\overline{G}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Our assumption that $F$ preserves $\kappa $-small limits guarantees that $F \circ \overline{G}$ is a limit diagram in $\operatorname{\mathcal{D}}$: that is, the restriction functor $\operatorname{\mathcal{D}}_{ / (F \circ \overline{G} )} \rightarrow \operatorname{\mathcal{D}}_{ / (F \circ G)}$ is a trivial Kan fibration. In particular, we can lift $D'$ to an object $\overline{D}' \in \operatorname{\mathcal{D}}_{ / (F \circ \overline{G} )}$. The pair $(\overline{G}, \overline{D}' )$ can then determines an extension of $\widetilde{G}$ to a diagram $K^{\triangleleft } \rightarrow \widetilde{\operatorname{\mathcal{C}}}$. $\square$