Kerodon

Remark 6.2.1.18. We will see later that the converse of Proposition 6.2.1.17 also holds: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty $-categories and $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is a natural transformation which induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D))$ for every pair of objects $(C,D) \in \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$, then $\eta $ is the unit of an adjunction between $F$ and $G$ (Corollary 6.2.4.5).