Corollary 15.1.4 (Contravariant left-adjoint unfurling). Let \((C,C,C_R)\) be an adequate triple. Every left \(C_R\)-adjointable functor \(F\colon C\catop \to \Cat _{\infty }\) extends canonically to a functor \[ \Unf ^{\mathrm {L}}(F)\colon \Span _{\all ,R}(C)\longrightarrow \Cat _{\infty } \] 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. Let \(p\colon \Un ^{\ct }(F)\to C\) be the cartesian unstraightening. The dual of Lemma 23.1.20, applied over each \(r\in C_R\), identifies the existence of \(r_!\) with that of \(p\)-cocartesian lifts over \(r\). The Beck–Chevalley condition for \(F\) is condition (3) of Definition 15.1.2, so Theorem 15.1.3 gives the extension and its transport formula. The descriptions of the fibers and backwards transport identify its restriction with \(F\). □
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