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 Proposition 6.3.3.13, 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. We will complete the proof by showing that $\eta $ is also an isomorphism. Fix an object $C \in \operatorname{\mathcal{C}}$, we wish to show that the map $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ (Theorem 4.4.4.4). By virtue of assumption $(1)$, it will suffice to show that $F( \eta _{C} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. The compatibility of $\eta $ and $\epsilon $ guarantees that the diagram

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$. Since $\epsilon _{F(C)}$ is an isomorphism, it follows that $F( \eta _ C )$ is also an isomorphism (Remark 1.3.6.3). $\square$