Kerodon

Remark 6.2.1.4. Let $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ be the homotopy $2$-category of $\infty $-categories (see Construction 4.5.1.23). Suppose we are given functors of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, which we regard as $1$-morphisms in the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be natural transformations and let $[\eta ]$ and $[\epsilon ]$ denote their homotopy classes, which we regard as $2$-morphisms in $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$. Then $\eta $ and $\epsilon $ are compatible up to homotopy (in the sense of Definition 6.2.1.1) if and only if the pair $( [\eta ], [\epsilon ])$ is an adjunction in the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ (in the sense of Definition 6.1.1.1).