Kerodon

Notation 6.1.3.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $C$ and $D$, and let $\underline{\operatorname{RHom}}_{\operatorname{\mathcal{C}}}(D,C)$ denote the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$ spanned by those $1$-morphisms $g: D \rightarrow C$ which admit a left adjoint $f: C \rightarrow D$. In this case, Corollary 6.1.3.3 guarantees that the $1$-morphism $f$ is uniquely determined up to isomorphism. We will sometimes abuse terminology by referring to $f$ as the left adjoint of $g$ and denoting it by $g^{L}$. The construction $g \mapsto g^{L}$ determines an equivalence of categories $\underline{\operatorname{RHom}}_{\operatorname{\mathcal{C}}}(D,C) \rightarrow \underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}}$, which is homotopy inverse to the functor $f \mapsto f^{R}$ described in Notation 6.1.3.5.