Kerodon

Notation 6.1.3.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $C$ and $D$, and let $\underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)$ denote the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)$ spanned by those $1$-morphisms $f: C \rightarrow D$ which admit a right adjoint $g: D \rightarrow C$. In this case, Corollary 6.1.3.3 guarantees that the $1$-morphism $g$ is determined uniquely up to isomorphism. We will sometimes abuse terminology by referring to $g$ as the right adjoint of $f$ and denoting it by $f^{R}$. The construction $f \mapsto f^{R}$ extends to a functor of categories $\underline{\operatorname{LHom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)$, which carries each $2$-morphism $\beta : f \Rightarrow f'$ to the $2$-morphism $\beta ^{R}: f'^{R} \Rightarrow f^{R}$ described in Proposition 6.1.3.4.