Kerodon

Proof. Let us abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ with full subcategories of the $\infty $-categories $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\widehat{\operatorname{\mathcal{D}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, respectively. Setting $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$, we can assume without loss of generality that $F = \widehat{F}|_{\operatorname{\mathcal{C}}}$. Fix a regular cardinal $\mu \geq \lambda $ such that $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ are $\mu $-small. By virtue of Proposition 7.4.3.11, it will suffice to show that for every functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ which is corepresentable by an object $D \in \operatorname{\mathcal{D}}$, the composite functor $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ has a contractible colimit. Let $\widehat{G}: \widehat{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ be the functor corepresented by $D$. Since $D$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{D}}}$, the functor $\widehat{G}$ is $(\kappa ,\lambda )$-finitary. It follows that the composite functor $( \widehat{G} \circ \widehat{F} ): \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is also $(\kappa ,\lambda )$-finitary, and is therefore left Kan extended from $\operatorname{\mathcal{C}}$ (Remark 9.3.1.14). Applying Corollary 7.3.8.3, we are reduced to showing that the colimit $\varinjlim ( \widehat{G} \circ \widehat{F} )$ a contractible Kan complex, which follows from our assumption that $\widehat{F}$ is right cofinal (Proposition 7.4.3.11). $\square$