Kerodon

Proof. Suppose that conditions $(1)$ and $(2)$ are satisfied; we will show that $F$ is an equivalence of $\infty $-categories (the converse is immediate from the definitions). Combining assumption $(2)$ with Corollary 6.2.2.17, we can choose a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation which is compatible up to homotopy with $\epsilon $, in the sense of Definition 6.2.1.1. For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\eta _{C}$ fits into a commutative diagram

in the $\infty $-category $\operatorname{\mathcal{D}}$, where $\epsilon _{ F(C) }$ and $\operatorname{id}_{ F(C) }$ are isomorphisms. It follows that $F( \eta _ C )$ is also an isomorphism in $\operatorname{\mathcal{D}}$. Applying assumption $(1)$, we deduce that $\eta _{C}$ is an isomorphism in $\operatorname{\mathcal{C}}$. Allowing the object $C$ to vary (and invoking the criterion of Theorem 4.4.4.4), we deduce that $\eta $ is also a natural isomorphism, so that $F$ and $G$ are homotopy inverse to one another. $\square$