Composition of right adjoints under straightening

Remark 23.2.12. Recall that if two functors \(F\colon C \to D\) and \(G\colon D \to E\) admit right adjoints \(F^R\) and \(G^R\), then \(GF\) admits a right adjoint \(F^RG^R\). Using straightening/unstraightening, we can define a highly coherent version of this functoriality of passing to right adjoints. Consider the wide subcategories \[ \CatL _{\infty } \subseteq \Cat _{\infty } \qquadtext { and } \CatR _{\infty } \subseteq \Cat _{\infty } \] denote the wide subcategories spanned by the left adjoints and right adjoints, respectively. We claim that passing to right adjoints defines a functor \[ (-)^{\R } \colon (\CatL _{\infty })\catop \to \CatR _{\infty }. \] To this end, choose a bigger universe \(\widehat {\Cat }_{\infty }\) containing \(\Cat _{\infty }\) as an object. The inclusion \(\CatL _{\infty } \hookrightarrow \widehat {\Cat }_{\infty }\) can then be straightened to a cocartesian fibration \(E \to \CatL _{\infty }\). Given a morphism \(F\colon C \to D\) in \(\CatL _{\infty }\), the induced cocartesian transport map \(E_C \to E_D\) on fibers is precisely the functor \(F\) itself. Since by assumption \(F\) admits a right adjoint, it follows from Lemma 23.1.20 that this cocartesian fibration is also a cartesian fibration, hence straightens to a functor \((-)^{\textup {R}} \colon (\CatL _{\infty })\catop \to \widehat {\Cat }_{\infty }\). By checking on objects and morphisms, this factors through \(\CatR _{\infty }\).