Kerodon

Proof. We will show that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$; the assertion that $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ follows by a similar argument. Choose a regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Using Corollary 8.3.3.17, we can choose a fully faithful functor $\operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where $\widehat{\operatorname{\mathcal{D}}}$ admits $\kappa $-small colimits. By virtue of Remark 7.3.1.13, we can replace $\operatorname{\mathcal{D}}$ by $\widehat{\operatorname{\mathcal{D}}}$ and thereby reduce to proving Proposition 8.5.1.8 in the special case where $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits.

Set $F^{0} = F|_{ \operatorname{\mathcal{C}}^{0} }$. Using Proposition 7.6.7.13, we can extend $F^{0}$ to a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Invoking the universal mapping property of Corollary 7.3.6.9, we deduce that there is a natural transformation $\alpha : F' \rightarrow F$ which restricts to the identity transformation from $F^{0}$ to itself. The natural transformation $\alpha $ carries each object $Y \in \operatorname{\mathcal{C}}$ to a morphism $\alpha _{Y}: F'(Y) \rightarrow F(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. By assumption, the object $Y$ is a retract of some object $X \in \operatorname{\mathcal{C}}^{0}$. It follows that $\alpha _ Y$ is a retract of the morphism $\alpha _{X} = \operatorname{id}_{ F(X) }$, and is therefore an isomorphism (Proposition 8.5.1.7). Invoking Theorem 4.4.4.4, we deduce that the natural transformation $\alpha $ is an isomorphism, so that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ by virtue of Remark 7.3.3.17. $\square$