Corollary 7.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a coreflective subcategory. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$.
- $(2)$
Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$ (Definition 6.2.2.1). Then $F(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.
- $(3)$
Let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a right adjoint to the inclusion. If $e$ is a morphism in $\operatorname{\mathcal{C}}$ and $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}_0$, then $F(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.