Kerodon

Remark 6.2.1.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Proposition 6.1.3.4 that if $F$ admits a right adjoint $G$, then $G$ is well-defined up to isomorphism as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$. We will sometimes emphasize this by referring to $G$ as the right adjoint of $F$ and denoting it by $F^{R}$. By virtue of Notation 6.1.3.8, the construction $F \mapsto F^{R}$ determines an equivalence of homotopy categories $\mathrm{h} \mathit{\operatorname{LFun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} \rightarrow \mathrm{h} \mathit{\operatorname{RFun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})}^{\operatorname{op}}$. We will see later that this construction can be upgraded to an equivalence of $\infty $-categories $\operatorname{LFun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{RFun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\operatorname{op}}$ (see Corollary 8.3.4.10).