Corollary 15.1.5 (Contravariant right-adjoint unfurling). Let \((C,C,C_R)\) be an adequate triple. Every right \(C_R\)-adjointable functor \(F\colon C\catop \to \Cat _{\infty }\) extends canonically to a functor \[ \Unf ^{\mathrm {R}}(F)\colon \Span _{\all ,R}(C)\longrightarrow \Cat _{\infty } \] which sends a span \(X\xleftarrow {l}U\xrightarrow {r}Y\) to \(r_*l^*\). Its restriction along \(C\catop \hookrightarrow \Span _{\all ,R}(C)\) is \(F\) itself.
Proof. The pointwise opposite \(F^{\mathrm {op}}\colon C\catop \to \Cat _{\infty }\) is left \(C_R\)-adjointable: the left adjoint to \((r^*)^{\mathrm {op}}\) is \((r_*)^{\mathrm {op}}\), and its Beck–Chevalley transformations are the opposites of those for \(F\). Apply Corollary 15.1.4 to \(F^{\mathrm {op}}\) and take opposites pointwise once more; the transport functor becomes \(r_*l^*\). □
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