Kerodon

Remark 6.2.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. It follows from Remark 6.2.1.4 that the condition that $\eta $ is the unit of an adjunction (in the sense of Definition 6.2.1.1) depends only on the homotopy class $[\eta ]$, regarded as a morphism in the category $\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})}$. Moreover, if $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is a counit which is compatible with $\eta $ up to homotopy, then the homotopy class $[\epsilon ]$ is uniquely determined (see Proposition 6.1.2.9). Beware that it is only the homotopy class of $\epsilon $ that is uniquely determined: if $\epsilon ': F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is homotopic to $\epsilon $, then it is also compatible with $\eta $ up to homotopy.